Chapter 8.3, Problem 5P

a.

To determine

To find: The simpler form of $\left(2x^2-3x+1\right)\left(x-3\right)$

Answer to Problem 5P

The simpler form of expression $\left(2x^2-3x+1\right)\left(x-3\right)$ is $2x^3-9x^2+10x-3$ .

Explanation of Solution

Given:

The expression for finding simpler form is $\left(2x^2-3x+1\right)\left(x-3\right)$

Formula/ concept used:

The simpler form of the expression as a product of factors is an algebraic expression obtained by multiplying its all factors.

Calculations:

The simpler form of expression $\left(2x^2-3x+1\right)\left(x-3\right)$ is obtained as polynomial of degree 3 obtained by multiplying the quadratic function $\left(2x^2-3x+1\right)$ and a linear function $\left(x-3\right)$

Now, $\left(2x^2-3x+1\right)\left(x-3\right)=\left(2x^2-3x+1\right)\cdot x+\left(2x^2-3x+1\right)\cdot \left(-3\right)$

  $\Rightarrow \left(2x^2-3x+1\right)\cdot x-\left(2x^2-3x+1\right)\cdot 3$

  $\Rightarrow \left(2x^3-3x^2+x\right)-\left(6x^2-9x+3\right)$

  $\Rightarrow 2x^3-3x^2+x-6x^2+9x-3$

  $\Rightarrow 2x^3-9x^2+10x-3$ .

Thus, the simpler form of expression $\left(2x^2-3x+1\right)\left(x-3\right)$ is $2x^3-9x^2+10x-3$ .

Conclusion:

The simpler form of expression $\left(2x^2-3x+1\right)\left(x-3\right)$ is $2x^3-9x^2+10x-3$ .

b.

To determine

To find: The product of a trinomial and binomial using Distributive property.

Answer to Problem 5P

The product of a trinomial and binomial using Distributive property is $\left(2x^2-3x+1\right)\left(x-3\right)=\left(2x^2-3x+1\right)\cdot x+\left(2x^2-3x+1\right)\cdot \left(-3\right)=2x^3-9x^2+10x-3$

Explanation of Solution

Given:

Thegiven trinomial is $\left(2x^2-3x+1\right)$ and binomial $\left(x-3\right)$ .

Concept used:

The Distributive property of multiplication over addition states $a\left(b+c\right)=a\cdot b+a\cdot c$ .

Calculations:

Now, $\left(2x^2-3x+1\right)\left(x-3\right)=\left(2x^2-3x+1\right)\cdot x+\left(2x^2-3x+1\right)\cdot \left(-3\right)$

  $\Rightarrow \left(2x^2-3x+1\right)\cdot x-\left(2x^2-3x+1\right)\cdot 3$

  $\Rightarrow \left(2x^3-3x^2+x\right)-\left(6x^2-9x+3\right)$

  $\Rightarrow 2x^3-3x^2+x-6x^2+9x-3$

  $\Rightarrow 2x^3-9x^2+10x-3$ .

That was to show.

The product of a trinomial and binomial using Distributive property is $\left(2x^2-3x+1\right)\left(x-3\right)=\left(2x^2-3x+1\right)\cdot x+\left(2x^2-3x+1\right)\cdot \left(-3\right)=2x^3-9x^2+10x-3$