Question

Consider the polynomial$2x^5+6x^4+4x^3$

(a) How many terms does this polynomial have? List the terms.

(b) What factor is common to each term? Factor the polynomial.

Short answer

1. There are 3 terms in the polynomial $2x^5+6x^4+4x^3$which are $2x^5,6x^4,4x^3$
2. The common factor to each term is $2x^3$and the polynomial after factorization is$2x^5+6x^4+4x^3=2x^3\left(x^2+3x+2\right)$

Step-by-step solution

Part a. Step 1. Apply the concept of polynomials.

Apolynomial in the variable x is an expression of the form $a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\mathrm{...}+a_{1}x+a_0$

$a_n,a_{n-1},a_{n-2},\mathrm{...},a_1,a_0$ are real numbers, and n is a non-negative integer. If $a_n\ne 0$, then the polynomial hasdegree n. The monomials$a_k{x}^k$ that make up the polynomial are called theterms of the polynomial.

Part a. Step 2. Analyze the polynomial.

Given$2x^5+6x^4+4x^3$

We arrange the terms of the polynomial such the exponents are in descending order.

$2x^5+6x^4+4x^3$

Part a. Step 3. Find the number of terms of polynomial.

Clearly, there are 3 terms in the polynomial$2x^5+6x^4+4x^3$

Part a. Step 4. List the terms of polynomial.

The terms in the polynomial$2x^5+6x^4+4x^3$ are$2x^5,6x^4,4x^3$

Part b. Step 1. Apply the concept of polynomials.

Apolynomial in the variable x is an expression of the form $a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\mathrm{...}+a_{1}x+a_0$

$a_n,a_{n-1},a_{n-2},\mathrm{...},a_1,a_0$ are real numbers, and n is a non-negative integer. If $a_n\ne 0$, then the polynomial hasdegree n. The monomials$a_k{x}^k$ that make up the polynomial are called theterms of the polynomial.

Part b. Step 2. Analyze the polynomial.

Given$2x^5+6x^4+4x^3$

We arrange the terms of the polynomial such the exponents are in descending order.

$2x^5+6x^4+4x^3$

Part b. Step 3. Find the greatest common factor.

$2x^5+6x^4+4x^3$is the polynomial

2,6,4 have greatest common factor 2

$x^5,x^4,x^3$have greatest common factor $x^3$

So, the greatest common factor of the three terms is$2x^3$

Part b. Step 4. Factor the polynomial.

Now, the factors of polynomial $2x^5+6x^4+4x^3$are given by:

$2x^5+6x^4+4x^3=2x^3\left(x^2+3x+2\right)$