What is the degree of P(x) = 2x - 2?

The highest power of x is 1. Degree = 1.

What is the degree of P(x) = 2(x - 2)(x^2 + 5)?

Add the degrees of the factors: 1 + 2 = 3. Degree = 3.

What is the degree of P(x) = -5(x - 2)(x^3 + 5) + x^5?

The factored portion has degree 4; the standalone term x^5 dominates. Degree = 5.

Give the zeros and multiplicities of P(x) = (x + 2)(x - 1).

x = -2 (multiplicity 1) and x = 1 (multiplicity 1).

Give the zeros and multiplicities of P(x) = (x - 2)^2(x + 4).

x = 2 (multiplicity 2) and x = -4 (multiplicity 1).

Give the zeros and multiplicities of P(x) = (x + 5)^3(x - 6)(-x + 6).

Rewrite (-x + 6) = -(x - 6), giving P(x) = -(x + 5)^3 (x - 6)^2. Zeros: x = -5 (multiplicity 3), x = 6 (multiplicity 2).

Give the zeros and multiplicities of P(x) = (x + 1)(x + 2) + (x + 1)(x + 2)^2.

Factor GCF: P(x) = (x + 1)(x + 2)(x + 3). Zeros: x = -1, -2, -3, each with multiplicity 1.

Give the zeros and multiplicities of P(x) = (x - 4)(x^2 + 4)^2.

Real: x = 4 (multiplicity 1). Complex: x = 2i (multiplicity 2), x = -2i (multiplicity 2).

Give the zeros and multiplicities of P(x) = (-x + 3)(x^2 + 2x + 5)^2.

Real: x = 3 (multiplicity 1). Complex: x = -1 + 2i (multiplicity 2), x = -1 - 2i (multiplicity 2).

Give the zeros and multiplicities of P(x) = -x(-x - 2)^2(x + 2).

Rewrite (-x - 2)^2 = (x + 2)^2, giving P(x) = -x(x + 2)^3. Zeros: x = 0 (multiplicity 1), x = -2 (multiplicity 3).

Polynomial Degree and Zeros — Practice Questions

This page presents 10 practice questions on the degree of a polynomial and on finding zeros (roots) alongside their multiplicities. Problems include both real and complex roots. Complete step-by-step solutions are provided beneath each question — ideal for algebra and precalculus students.

Question 1 — Degree of a Linear Polynomial

What is the degree of the polynomial \( P(x) = 2x - 2 \)?

View Solution

The degree is the highest power of \(x\). Here \(P(x) = 2x^1 - 2\), so the highest power is 1.

Answer: Degree = 1

Question 2 — Degree of a Factored Polynomial

What is the degree of the polynomial \( P(x) = 2(x - 2)(x^2 + 5) \)?

View Solution

In factored form, add the degrees of the factors: \((x-2)\) has degree 1, \((x^2+5)\) has degree 2.

\[ 1 + 2 = 3 \]

Check by multiplying leading terms: \(x \cdot x^2 = x^3\).

Answer: Degree = 3

Question 3 — Degree When Two Terms Compete

What is the degree of the polynomial \( P(x) = -5(x - 2)(x^3 + 5) + x^5 \)?

View Solution

Leading term from the factored part:

\[ -5 \cdot x \cdot x^3 = -5x^4 \]

So \(P(x) = -5x^4 + \cdots + x^5\).

The two highest-degree terms have degrees 4 and 5 — since they are different, they cannot cancel. The \(x^5\) term dominates.

Answer: Degree = 5

Question 4 — Zeros of Two Linear Factors

Give the zeros of \( P(x) = (x + 2)(x - 1) \) and list their multiplicities.

View Solution

\(x + 2 = 0 \implies x = -2\) (exponent 1)
\(x - 1 = 0 \implies x = 1\) (exponent 1)

Answer: \(x = -2\) (multiplicity 1), \(x = 1\) (multiplicity 1)

Question 5 — Zeros with a Repeated Factor

Give the zeros of \( P(x) = (x - 2)^2(x + 4) \) and list their multiplicities.

View Solution

\((x - 2)^2 = 0 \implies x = 2\), multiplicity 2
\((x + 4)^1 = 0 \implies x = -4\), multiplicity 1

Answer: \(x = 2\) (multiplicity 2), \(x = -4\) (multiplicity 1)

Question 6 — Two Factors That Represent the Same Zero

Give the zeros of \( P(x) = (x + 5)^3(x - 6)(-x + 6) \) and list their multiplicities.

View Solution

Factor out \(-1\): \((-x + 6) = -(x - 6)\). Then

\[ P(x) = (x+5)^3 (x-6) \cdot [-(x-6)] = -(x+5)^3 (x-6)^2 \]

\(x + 5 = 0 \implies x = -5\), multiplicity 3
\(x - 6 = 0 \implies x = 6\), multiplicity 2

Answer: \(x = -5\) (multiplicity 3), \(x = 6\) (multiplicity 2)

Question 7 — Sum of Two Products with a Common Factor

Give the zeros of \( P(x) = (x + 1)(x + 2) + (x + 1)(x + 2)^2 \) and list their multiplicities.

View Solution

Factor out the GCF \((x+1)(x+2)\):

\[ P(x) = (x+1)(x+2)\left[1 + (x+2)\right] = (x+1)(x+2)(x+3) \]

\(x = -1\), multiplicity 1
\(x = -2\), multiplicity 1
\(x = -3\), multiplicity 1

Answer: \(x = -1, -2, -3\), each with multiplicity 1

Question 8 — Linear Factor × Squared Irreducible Quadratic

Give the zeros of \( P(x) = (x - 4)(x^2 + 4)^2 \) and list their multiplicities.

View Solution

Real zero: \(x - 4 = 0 \implies x = 4\), multiplicity 1.

Complex zeros: \(x^2 + 4 = 0 \implies x^2 = -4 \implies x = \pm 2i\).

The quadratic factor is squared, so each complex root inherits multiplicity 2.

Answer: \(x = 4\) (m=1), \(x = 2i\) (m=2), \(x = -2i\) (m=2)

Question 9 — Complex Zeros from the Quadratic Formula

Give the zeros of \( P(x) = (-x + 3)(x^2 + 2x + 5)^2 \) and list their multiplicities.

View Solution

Real zero: \(-x + 3 = 0 \implies x = 3\), multiplicity 1.

Complex zeros: Solve \(x^2 + 2x + 5 = 0\) with the quadratic formula:

\[ x = \frac{-2 \pm \sqrt{4 - 20}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i \]

The quadratic is squared, so each complex root has multiplicity 2.

Answer: \(x = 3\) (m=1), \(x = -1 + 2i\) (m=2), \(x = -1 - 2i\) (m=2)

Question 10 — Combining Factors into a Cube

Give the zeros of \( P(x) = -x(-x - 2)^2(x + 2) \) and list their multiplicities.

View Solution

Simplify \((-x-2)^2\) by pulling out \(-1\):

\[ (-x-2)^2 = [-(x+2)]^2 = (x+2)^2 \]

Substitute:

\[ P(x) = -x(x+2)^2(x+2) = -x(x+2)^3 \]

\(-x = 0 \implies x = 0\), multiplicity 1
\((x+2)^3 = 0 \implies x = -2\), multiplicity 3

Answer: \(x = 0\) (m=1), \(x = -2\) (m=3)

Frequently Asked Questions

How do I find the degree of a polynomial in factored form?

Add the degrees of each factor. For example, \((x-2)(x^2+5)\) has degree \(1 + 2 = 3\).

What is the multiplicity of a zero?

The multiplicity is the exponent of the factor that produces the zero. For \((x-2)^3\), the zero \(x=2\) has multiplicity 3.

Can a polynomial have complex zeros?

Yes. If an irreducible quadratic factor like \(x^2 + 4\) appears, it produces complex-conjugate zeros such as \(x = \pm 2i\).

Why do complex zeros come in conjugate pairs?

If a polynomial has real coefficients, then whenever \(a + bi\) is a zero, \(a - bi\) must also be a zero. This is the Complex Conjugate Root Theorem.

What does the highest-degree term tell me about the polynomial?

It determines the polynomial's end behavior and its maximum possible number of real zeros (counting multiplicity).