What is a zero of a polynomial?

A zero (also called a root) of p(x) is a value of x that makes p(x) = 0. Real zeros are the x-intercepts of the graph. Example: p(x) = x³ − 15x² + 71x − 105 factors as (x − 3)(x − 5)(x − 7), so its zeros are 3, 5, 7 — the graph crosses the x-axis at each one.

How do you find the zeros of a polynomial?

Set p(x) = 0, then: (1) factor out any common factor such as x or x²; (2) for a quadratic, factor it or use the quadratic formula; (3) for degree 3 and up, list rational candidates ±p/q (factors of the constant over factors of the leading coefficient), test them with synthetic division, and divide out each zero you find; (4) solve the remaining quadratic; (5) if no rational zero exists, use a numerical method. The calculator above does all five and shows each step.

If 1 and −3 are zeros of p(x) = x³ − ax² − 13x + b, find a and b.

a = 3 and b = 15. Substitute each zero: p(1) = 1 − a − 13 + b = 0 gives b − a = 12; p(−3) = −27 − 9a + 39 + b = 0 gives b − 9a = −12. Subtracting, 8a = 24, so a = 3 and b = 15. Then p(x) = x³ − 3x² − 13x + 15 = (x + 3)(x − 1)(x − 5), so the third zero is 5.

Find the zeros of 3x² + x − 10 and verify the relationship between the zeros and the coefficients.

3x² + x − 10 = (x + 2)(3x − 5), so the zeros are −2, 5/3. Check: sum of zeros = −2 + 5/3 = −1/3 = −b/a = −1/3, and product = −2 × 5/3 = −10/3 = c/a = −10/3. Both relationships hold.

How do I find the zeros of a quadratic that does not factor, such as 2x² − 10x − 3?

Use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. Here a = 2, b = −10, c = −3, so b² − 4ac = 100 + 24 = 124 and x = (10 ± √124)/4 = (5 − √31)/2 and (5 + √31)/2 ≈ −0.283882 and 5.28388.

Find the zeros of 2x² − 6.9x + 1 to the nearest thousandth.

Multiply by 10 to clear the decimal: 20x² − 69x + 10 = 0. The quadratic formula gives x = (69 − √3961)/40 and (69 + √3961)/40, which round to 0.152 and 3.298.

What are the zeros of f(x) = x(x² − 36)?

Set each factor to zero: x = 0, and x² − 36 = (x + 6)(x − 6) = 0 gives x = ±6. So f(x) = x(x + 6)(x − 6) has zeros −6, 0, 6. Similarly x³ − 9x² − 36x = x(x + 3)(x − 12) has zeros −3, 0, 12.

How do you find zeros and their multiplicity, e.g. f(x) = 4(x + 3)²(x − 1)?

The exponent on each factor is the multiplicity. f(x) = 4(x + 3)²(x − 1) has zero −3 with multiplicity 2 (the graph touches the x-axis and turns around) and zero 1 with multiplicity 1 (the graph crosses). The leading coefficient 4 does not create a zero. Another example: x⁵ + 10x⁴ + 25x³ = x³(x + 5)² has 0 with multiplicity 3 and −5 with multiplicity 2.

How do I find a polynomial with given zeros?

Multiply one factor (x − zero) for each zero, then by the leading coefficient. Zeros −2 and 5 give (x + 2)(x − 5) = x² − 3x − 10; zeros −3 and 4 give x² − x − 12; zeros 3 and 6 give x² − 9x + 18. With zeros −2, 1, 3 and leading coefficient 2: 2(x + 2)(x − 1)(x − 3) = 2x³ − 4x² − 10x + 12. Use the "Polynomial from zeros" tab to build any of these.

If 1 is a zero of x⁴ − 4x² − 7x + 10, how do you find all the zeros?

Divide by (x − 1) using synthetic division: x⁴ − 4x² − 7x + 10 = (x − 1)(x³ + x² − 3x − 10). The cubic has no rational zeros (±1, ±2, ±5, ±10 all fail), so its zeros are found numerically: one real zero ≈ 2.26765 and a complex pair ≈ −1.63382 ± 1.31927i. All four zeros: 1, 2.26765, −1.63382 − 1.31927i, −1.63382 + 1.31927i.

If x = 4/3 is a zero of 6x³ − 11x² + kx − 20, find k.

Substitute x = 4/3: 6(64/27) − 11(16/9) + 4k/3 − 20 = 0 → 128/9 − 176/9 + 4k/3 − 20 = 0 → 4k/3 = 20 + 48/9 = 76/3 → k = 19. Check: p(4/3) = 0, and 6x³ − 11x² + 19x − 20 = (3x − 4)(2x² − x + 5), so the other zeros are complex: (1 − i√39)/4 and (1 + i√39)/4.

How do you find the zero of a linear polynomial like 5x − 20 or 2y + 7?

Set it equal to zero and solve: ax + b = 0 gives x = −b/a. 5x − 20 → 4; 2y + 7 → −7/2; 3x − 4 → 4/3; −5x + 50 → 10. A linear polynomial always has exactly one zero.

How many zeros can a polynomial have?

A polynomial of degree n has exactly n zeros counted with multiplicity, if complex zeros are included (Fundamental Theorem of Algebra). It has at most n real zeros. Complex zeros of real polynomials come in conjugate pairs, so an odd-degree polynomial always has at least one real zero. Descartes' rule of signs bounds how many are positive or negative: for −x⁴ + 3x³ + 9x² − 23x + 12, the calculator finds −3, 1 (multiplicity 2), 4 — 4 zeros counting multiplicity.

What are complex zeros and how do you find them?

Complex zeros appear when the discriminant of a quadratic factor is negative. For x² − 3x + 3, b² − 4ac = 9 − 12 = −3, so x = (3 ± i√3)/2 ≈ 1.5 − 0.866025i and 1.5 + 0.866025i. The graph never touches the x-axis. Complex zeros always come in conjugate pairs a ± bi when the coefficients are real.

Are zeros, roots and x-intercepts the same thing?

Nearly. Zeros belong to a function (values of x where f(x) = 0), roots belong to an equation (solutions of p(x) = 0), and x-intercepts are points on a graph (x, 0). For real zeros they give the same numbers. Complex zeros are zeros and roots but not x-intercepts. "Zeros" and "zeroes" are both accepted spellings.

How do you find the zeros and holes of a rational function?

Factor the numerator and denominator. Factors that cancel give holes; the remaining numerator zeros are the zeros of the function; the remaining denominator zeros are vertical asymptotes. For (x² − 4) / (x² − x − 2): zero −2, hole at (2, 4/3), vertical asymptote x = −1. For (2x − 5) / (2x − 1): zero x = 5/2, vertical asymptote x = 1/2, horizontal asymptote y = 1, no holes.

What is the rational root theorem?

If a polynomial has integer coefficients, every rational zero p/q (in lowest terms) has p dividing the constant term and q dividing the leading coefficient. For −x⁴ + 3x³ + 9x² − 23x + 12, the candidates are ±1, ±2, ±3, ±4, ±6, ±12; testing them finds −3, 1 (multiplicity 2), 4, so it factors as −(x + 3)(x − 1)²(x − 4).

Is f(0) the same as a zero of the function?

No. f(0) is the y-intercept — the output when x = 0. A zero is an input x where the output is 0. For x³ − 15x² + 71x − 105, f(0) = −105 but the zeros are 3, 5, 7. x = 0 is a zero only when the constant term is 0, as in x² + 11x = x(x + 11).

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Q: If the zeros of x³ − 15x² + 71x − k are in arithmetic progression, what is k?

k = 105. Write the zeros as a − d, a, a + d. Their sum is 3a = 15, so a = 5. The sum of products of pairs is 3a² − d² = 71, so 75 − d² = 71 and d = 2. The zeros are 3, 5, 7, and k = product of zeros = 3 × 5 × 7 = 105. Check: (x − 3)(x − 5)(x − 7) = x³ − 15x² + 71x − 105.

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Real, repeated & complex zeros · step by step

Zeros Calculator — Find the Zeros of a Polynomial

Quick answer

The zeros of a polynomial are the x-values where p(x) = 0. To find them, factor out common terms, test rational candidates ±(factors of the constant)/(factors of the leading coefficient) with synthetic division, then solve the leftover quadratic with the quadratic formula. A degree-n polynomial has exactly n zeros counting multiplicity and complex zeros. Example: x³ − 15x² + 71x − 105 = (x − 3)(x − 5)(x − 7), so the zeros are 3, 5, 7.

Type any polynomial — such as x^3 - 6x^2 + 11x - 6, 4(x+3)^2(x-1) or 2x² − 6.9x + 1 — or a rational function like (x^2-4)/(x^2-x-2). You get exact zeros, multiplicities, the factored form, a graph and every step. Switch tabs to build a polynomial from its zeros.

Exact rational, surd and complex zeros (e.g. (5 ± √31)/2, (3 ± i√3)/2)
Multiplicity and whether the graph crosses or touches
Rational root candidates, Descartes’ rule and Vieta checks
Zeros, holes and asymptotes of rational functions

Last updated October 1, 2026. Checked against OpenStax College Algebra and NCERT Class 10. · how to find zeros · polynomial from zeros

Zeros Calculator

All real and complex zeros, exact where possible, with multiplicity, factored form, graph and steps.

Polynomial or function f(x)

Use ^ for powers (x^3), implied multiplication 4(x+3)^2(x−1), fractions 1/2x², √3, decimals, or a fraction like (2x−5)/(2x−1).

Zeros of f(x) = x³ − 15x² + 71x − 105

x = 3x = 5x = 7

Factored: f(x) = (x − 3)(x − 5)(x − 7)

Zero	Decimal	Mult.	Type / graph
3	3	1	rational · crosses the x-axis
5	5	1	rational · crosses the x-axis
7	7	1	rational · crosses the x-axis

Degree · zeros

3 · 3 real, 0 complex

y-intercept

(0, −105)

End behavior

Falls left, rises right: f(x) → −∞ as x → −∞ and +∞ as x → +∞

Descartes’ rule of signs

Positive real zeros: 3 or 1 · negative: 0

Relationship between zeros and coefficients (verified)
Sum of zeros	15	−(−15)/1 = 15 ✓
Sum of products of pairs	71	71/1 = 71 ✓
Product of zeros	105	−(−105)/1 = 105 ✓

Step-by-step

Write in standard form. f(x) = x³ − 15x² + 71x − 105. Degree 3, so there are exactly 3 zeros counting multiplicity (some may be complex).
List possible rational zeros. Rational root theorem: any rational zero is ±p/q, with p a factor of the constant −105 and q a factor of the leading coefficient 1. Candidates: 1, −1, 3, −3, 5, −5, 7, −7, 15, −15, 21, −21, 35, −35, 105, −105.
Test x = 3: f(3) = 0 ✓. Divide x³ − 15x² + 71x − 105 by (x − 3) (synthetic division with 3): quotient x² − 12x + 35, remainder 0.
Factor the quadratic. x² − 12x + 35 = (x − 5)(x − 7) (the discriminant b² − 4ac = 4 is a perfect square, so it factors over the integers). Set each factor to 0: x = 5 or 7.
Answer. Zeros: x = 3; x = 5; x = 7. 3 real and 0 complex, counting multiplicity.

Worked examples: zeros of common polynomials

Every row below is computed by the same engine as the calculator.

Polynomial	Factored form	Zeros	Why it's useful
x³ − 15x² + 71x − 105	(x − 3)(x − 5)(x − 7)	3; 5; 7	Arithmetic-progression cubic (k = 105)
x³ − 3x² − 13x + 15	(x + 3)(x − 1)(x − 5)	−3; 1; 5	1 and −3 are zeros → a = 3, b = 15
3x² + x − 10	(x + 2)(3x − 5)	−2; 5/3	Verify zeros vs coefficients
x³ − 36x	x(x + 6)(x − 6)	−6; 0; 6	Factor out x first
4x³ + 20x² + 12x − 36	4(x + 3)²(x − 1)	−3 (multiplicity 2); 1	Zero with multiplicity 2
−x⁴ + 3x³ + 9x² − 23x + 12	−(x + 3)(x − 1)²(x − 4)	−3; 1 (multiplicity 2); 4	Repeated rational zero
x⁴ + x³ − 12x²	x²(x + 4)(x − 3)	−4; 0 (multiplicity 2); 3	x² factor + quadratic
x⁵ + 10x⁴ + 25x³	x³(x + 5)²	−5 (multiplicity 2); 0 (multiplicity 3)	x³(x + 5)²
2x² − 10x − 3	2x² − 10x − 3	(5 − √31)/2 ≈ −0.283882; (5 + √31)/2 ≈ 5.28388	Irrational zeros (surds)
2x² − 6.9x + 1	0.1(20x² − 69x + 10)	(69 − √3961)/40 ≈ 0.151588; (69 + √3961)/40 ≈ 3.29841	Decimal coefficients
x² − 3x + 3	x² − 3x + 3	(3 − i√3)/2 ≈ 1.5 − 0.866025i; (3 + i√3)/2 ≈ 1.5 + 0.866025i	Complex conjugate zeros
1.73205x² − 8x + 6.9282	√3(x − 2√3/3)(x − 2√3)	2√3/3 ≈ 1.1547; 2√3 ≈ 3.4641	Irrational coefficients

How to find the zeros of a polynomial

Finding zeros means solving p(x) = 0. Work from the easiest method to the hardest, and divide out every zero you find so the polynomial gets smaller.

1. Factor out the GCF
Take out common numbers and powers of x. x⁴ + x³ − 12x² = x²(x² + x − 12) = x²(x + 4)(x − 3), so 0 is a zero of multiplicity 2.
2. Factor or use the quadratic formula
For ax² + bx + c, try factoring first. If it does not factor, x = (−b ± √(b² − 4ac)) / 2a gives exact zeros.
3. List rational candidates
Rational root theorem: ±p/q with p | constant and q | leading coefficient. For x³ − 15x² + 71x − 105: 1, −1, 3, −3, 5, −5, 7, −7, 15, −15, ….
4. Test with synthetic division
A remainder of 0 means the candidate is a zero (factor theorem). Keep the quotient and divide again by the same zero to check multiplicity.
5. Solve what is left
Once the quotient is quadratic, factor it or use the formula. A negative discriminant gives a complex conjugate pair.
6. Use numbers when algebra stops
Cubic and higher factors with no rational zeros are solved numerically (this calculator uses Durand–Kerner with Newton polishing), e.g. (x − 1)(x³ + x² − 3x − 10) has a real zero ≈ 2.26765.

Method by degree

Which technique finds the zeros fastest

Degree	Method	Number of zeros
1 (linear)ax + b	x = −b/a	Exactly 1 real zero
2 (quadratic)ax² + bx + c	Factor, complete the square or quadratic formula	2 (real, repeated or a complex pair)
3 (cubic)ax³ + bx² + cx + d	Rational root theorem + synthetic division, then a quadratic	3; at least 1 real
4 (quartic)ax⁴ + … + e	GCF, substitution u = x² if biquadratic, rational roots, then quadratics	4; 0, 2 or 4 real
5+aₙxⁿ + … + a₀	Factor what you can; no general formula exists — use numerical methods	n; odd degree has at least 1 real

Discriminant and the type of zeros

For a quadratic ax² + bx + c

Discriminant	Zeros	Example
b² − 4ac > 0, perfect square	Two distinct rational zeros	x² − 2x − 15 = (x + 3)(x − 5) → −3, 5
b² − 4ac > 0, not a square	Two distinct irrational zeros	2x² − 10x − 3 → (5 − √31)/2, (5 + √31)/2
b² − 4ac = 0	One repeated zero (multiplicity 2)	x² − 6x + 9 = (x − 3)² → 3, 3
b² − 4ac < 0	Two complex conjugate zeros	x² − 3x + 3 → (3 − i√3)/2, (3 + i√3)/2

Zeros of x³ − 15x² + 71x − k in arithmetic progression

A classic exam problem: “If the zeros of x³ − 15x² + 71x − k are in arithmetic progression (AP), find k.” Let the zeros be a − d, a and a + d, then use the relationships between zeros and coefficients:

Sum of zeros: (a − d) + a + (a + d) = 3a = 15, so a = 5.
Sum of products of pairs: 3a² − d² = 71, so 75 − d² = 71, d² = 4 and d = 2.
Zeros: 3, 5, 7. Product of zeros = k = 3 × 5 × 7 = 105.
Check: (x − 3)(x − 5)(x − 7) = x³ − 15x² + 71x − 105. ✓

Cubic x³ − px² + qx − k	Middle zero a = p/3	d² = 3a² − q	Zeros	k = a(a² − d²)
x³ − 15x² + 71x − k	5	4	3, 5, 7	105
x³ − 6x² + 11x − k	2	1	1, 2, 3	6
x³ − 9x² + 23x − k	3	4	1, 3, 5	15
x³ − 12x² + 39x − k	4	9	1, 4, 7	28
x³ − 3x² + x − k	1	2	1 − √2, 1, 1 + √2	−1

Finding an unknown coefficient from a given zero

If r is a zero, then p(r) = 0 (factor theorem). Substitute each known zero and solve for the unknowns, then use the calculator to find the remaining zeros.

1 and −3 are zeros of x³ − ax² − 13x + b

p(1) = 1 − a − 13 + b = 0 → b − a = 12

p(−3) = −27 − 9a + 39 + b = 0 → b − 9a = −12

Subtract: 8a = 24 → a = 3, b = 15

x³ − 3x² − 13x + 15 = (x + 3)(x − 1)(x − 5) → zeros −3, 1, 5

x = 4/3 is a zero of 6x³ − 11x² + kx − 20

6(4/3)³ − 11(4/3)² + k(4/3) − 20 = 0

128/9 − 176/9 + 4k/3 − 20 = 0 → 4k/3 = 76/3 → k = 19

6x³ − 11x² + 19x − 20 = (3x − 4)(2x² − x + 5)

Other zeros: (1 − i√39)/4 and (1 + i√39)/4

Multiplicity and the graph

How a zero's multiplicity shapes the x-intercept

Multiplicity	Graph at the zero	Example
1 (odd)	Crosses the x-axis	x − 1 in 4(x + 3)²(x − 1)
2 (even)	Touches and turns around (bounces)	(x + 3)² in 4(x + 3)²(x − 1)
3 (odd)	Crosses and flattens (inflection at the axis)	x³ in x³(x + 5)²
Complex	No x-intercept	x² − 3x + 3

4x³ + 20x² + 12x − 36 = 4(x + 3)²(x − 1): zeros −3 (multiplicity 2), 1. Sum of multiplicities = degree = 3.

Relationship between zeros and coefficients

Vieta's formulas (NCERT Class 10, Chapter 2)

Polynomial	Sum	Pairs	Product
Quadratic ax² + bx + c	α + β = −b/a	—	αβ = c/a
Cubic ax³ + bx² + cx + d	α + β + γ = −b/a	αβ + βγ + γα = c/a	αβγ = −d/a
Quartic ax⁴ + bx³ + cx² + dx + e	−b/a	c/a	e/a

Check for 3x² + x − 10 (zeros −2, 5/3): sum of zeros −1/3 = −(1)/3 = −1/3; product of zeros −10/3 = (−10)/3 = −10/3.

Find a polynomial with given zeros

Reverse the process: p(x) = a(x − r₁)(x − r₂)…(x − rₙ). Pick the leading coefficient a (often 1), multiply out, and clear fractions if you want integer coefficients. Complex zeros need their conjugate for real coefficients.

Zeros	Factored form	Standard form
−2 and 5	(x + 2)(x − 5)	x² − 3x − 10
−3 and 4	(x + 3)(x − 4)	x² − x − 12
3 and 6	(x − 3)(x − 6)	x² − 9x + 18
3 and 5	(x − 3)(x − 5)	x² − 8x + 15
1/4 and −1	(x − 1/4)(x + 1)	x² + (3/4)x − 1/4integer form: 4x² + 3x − 1
−2, 1, 3 (leading coefficient 2)	2(x + 2)(x − 1)(x − 3)	2x³ − 4x² − 10x + 12
√2, −√2, 3	(x − √2)(x + √2)(x − 3)	x³ − 3x² − 2x + 6
2 + 3i (conjugate added)	(x − (2 + 3i))(x − (2 − 3i))	x² − 4x + 13

Zeros of rational functions: zeros vs holes vs asymptotes

For f(x) = N(x)/D(x), factor both parts. A factor that cancels makes a hole. The remaining zeros of N are the zeros of f; the remaining zeros of D are vertical asymptotes. Compare degrees for the horizontal asymptote.

Function	Zeros	Holes	Vertical asymptotes	Horizontal asymptote	y-intercept
(x² − 4) / (x² − x − 2)	x = −2	(2, 4/3)	x = −1	y = 1	2
(2x − 5) / (2x − 1)	x = 5/2	None	x = 1/2	y = 1	5

How to type a function

Powers: x^3, x³ or x3; multiplication can be implied: 4(x+3)^2(x-1)
Prefixes are ignored: f(x) =, p(y) =, y =, = 0
Square roots and decimals: √3x^2 - 8x + 4√3, sqrt(2)x, 2x^2 - 6.9x + 1
Rational functions: (x^2-4)/(x^2-x-2); any single letter works as the variable (x, y, t)

Sources

OpenStax College Algebra 2e — 5.5 Zeros of Polynomial FunctionsRational zero theorem, factor theorem, Descartes’ rule and the Fundamental Theorem of Algebra
OpenStax College Algebra 2e — 5.3 Graphs of Polynomial FunctionsMultiplicity and crossing vs touching the x-axis
Paul’s Online Math Notes — Finding Zeroes of PolynomialsWorked rational-root and synthetic division examples
Paul’s Online Math Notes — Zeroes/Roots of PolynomialsDefinitions of zeros, multiplicity and the factor theorem
NCERT Mathematics Class 10 — Chapter 2: Polynomials (PDF)Relationship between zeros and coefficients; source of many exam problems above
Wolfram MathWorld — Rational Zero TheoremStatement and proof sketch
Wolfram MathWorld — Descartes’ Sign RuleBounds on the number of positive and negative real zeros
Wolfram MathWorld — Fundamental Theorem of AlgebraA degree-n polynomial has exactly n complex zeros with multiplicity
Wikipedia — Vieta’s formulasSums and products of zeros for any degree
Wikipedia — Durand–Kerner methodThe simultaneous root-finding method used for zeros with no closed form

Need the division steps? Use the synthetic division calculator or the quadratic formula calculator.

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Zeros Calculator — Find the Zeros of a Polynomial

Worked examples: zeros of common polynomials

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Zeros of x³ − 15x² + 71x − k in arithmetic progression

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Find a polynomial with given zeros

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Sources

Frequently Asked Questions

What is a zero of a polynomial?

How do you find the zeros of a polynomial?

If the zeros of x³ − 15x² + 71x − k are in arithmetic progression, what is k?

If 1 and −3 are zeros of p(x) = x³ − ax² − 13x + b, find a and b.

Find the zeros of 3x² + x − 10 and verify the relationship between the zeros and the coefficients.

How do I find the zeros of a quadratic that does not factor, such as 2x² − 10x − 3?

Find the zeros of 2x² − 6.9x + 1 to the nearest thousandth.

What are the zeros of f(x) = x(x² − 36)?

How do you find zeros and their multiplicity, e.g. f(x) = 4(x + 3)²(x − 1)?

How do I find a polynomial with given zeros?

If 1 is a zero of x⁴ − 4x² − 7x + 10, how do you find all the zeros?

If x = 4/3 is a zero of 6x³ − 11x² + kx − 20, find k.

How do you find the zero of a linear polynomial like 5x − 20 or 2y + 7?

How many zeros can a polynomial have?

What are complex zeros and how do you find them?

Are zeros, roots and x-intercepts the same thing?

How do you find the zeros and holes of a rational function?

What is the rational root theorem?

Is f(0) the same as a zero of the function?

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Zeros Calculator — Find the Zeros of a Polynomial

Worked examples: zeros of common polynomials

How to find the zeros of a polynomial

Zeros of x³ − 15x² + 71x − k in arithmetic progression

Finding an unknown coefficient from a given zero

Find a polynomial with given zeros

More solved problems

Zeros of rational functions: zeros vs holes vs asymptotes

How to type a function

Sources

Frequently Asked Questions

What is a zero of a polynomial?

How do you find the zeros of a polynomial?

If the zeros of x³ − 15x² + 71x − k are in arithmetic progression, what is k?

If 1 and −3 are zeros of p(x) = x³ − ax² − 13x + b, find a and b.

Find the zeros of 3x² + x − 10 and verify the relationship between the zeros and the coefficients.

How do I find the zeros of a quadratic that does not factor, such as 2x² − 10x − 3?

Find the zeros of 2x² − 6.9x + 1 to the nearest thousandth.

What are the zeros of f(x) = x(x² − 36)?

How do you find zeros and their multiplicity, e.g. f(x) = 4(x + 3)²(x − 1)?

How do I find a polynomial with given zeros?

If 1 is a zero of x⁴ − 4x² − 7x + 10, how do you find all the zeros?

If x = 4/3 is a zero of 6x³ − 11x² + kx − 20, find k.

How do you find the zero of a linear polynomial like 5x − 20 or 2y + 7?

How many zeros can a polynomial have?

What are complex zeros and how do you find them?

Are zeros, roots and x-intercepts the same thing?

How do you find the zeros and holes of a rational function?

What is the rational root theorem?

Is f(0) the same as a zero of the function?

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