Chapter 1.3, Problem 122E

To determine

To calculate: The factor of the expression $y^4{\left(y+2\right)}^3+y^5{\left(y+2\right)}^4$ .

Answer to Problem 122E

The factor of the expression $y^4{\left(y+2\right)}^3+y^5{\left(y+2\right)}^4$ is $y^4{\left(y+2\right)}^3{\left(y+1\right)}^2$ .

Explanation of Solution

Given information:

The expression $y^4{\left(y+2\right)}^3+y^5{\left(y+2\right)}^4$ .

Formula used:

To factor out the common factor from a polynomial, find out the greatest common factor and express the polynomial as a product of the simpler ones.

The special factoring formula for perfect square, which is mathematically expressed as,

  $X^2+2XY+Y^2={\left(X+Y\right)}^2$

Calculation:

Consider the given expression $y^4{\left(y+2\right)}^3+y^5{\left(y+2\right)}^4$ .

Recall that to factor out the common factor from a polynomial, find out the greatest common factor and express the polynomial as a product of the simpler ones.

The common factor of these terms is $y^4{\left(y+2\right)}^3$ .

Simplify $y^4{\left(y+2\right)}^3+y^5{\left(y+2\right)}^4$ further as,

  $\begin{array}{c}{y}^4{\left(y+2\right)}^3+y^5{\left(y+2\right)}^4=y^4{\left(y+2\right)}^3\left[1+y\left(y+2\right)\right]\\ =y^4{\left(y+2\right)}^3\left(1+y^2+2y\right)\\ =y^4{\left(y+2\right)}^3\left(y^2+2y+1\right)\end{array}$

Recall the special factoring formula for perfect square, which is mathematically expressed as,

  $X^2+2XY+Y^2={\left(X+Y\right)}^2$

Apply it to factorize $\left(y^2+2y+1\right)$ , where $X=y\text{ and }Y=1$

  $\begin{array}{c}{y}^4{\left(y+2\right)}^3+y^5{\left(y+2\right)}^4=y^4{\left(y+2\right)}^3\left(y^2+2y+1\right)\\ =y^4{\left(y+2\right)}^3{\left(y+1\right)}^2\end{array}$

Thus, the factor of expression $y^4{\left(y+2\right)}^3+y^5{\left(y+2\right)}^4$ is $y^4{\left(y+2\right)}^3{\left(y+1\right)}^2$ .