Question

7 fill in the blank 1 point write the expression \\((x - 2)(x + 3)(x + 5)\\) as a polynomial in standard form. note: type your answer as is. ex. 2x^3 - x^2 + x - 7 for \\(2x^3 - x^2 + x - 7\\). do not use any spaces in your answer. type your answer... 8 multiple choice 1 point factor \\(x^3 - 5x^2 - 4x + 20\\) into linear factors. \\(\circ (x^2 - 4)(x + 5)\\) \\(\circ (x + 2)(x - 2)(x - 5)\\) \\(\circ (x^2 - 4)(x - 5)\\) \\(\circ (x + 2)(x - 2)(x + 5)\\)

Explanation:

Question 7

Step1: Multiply the first two factors

First, multiply \((x - 2)(x + 3)\). Using the distributive property (FOIL method):
\[

$$\begin{align*} (x - 2)(x + 3)&=x\cdot x + x\cdot 3 - 2\cdot x - 2\cdot 3\\ &=x^{2}+3x - 2x - 6\\ &=x^{2}+x - 6 \end{align*}$$

\]

Step2: Multiply the result by the third factor

Now, multiply \((x^{2}+x - 6)\) by \((x + 5)\):
\[

$$\begin{align*} (x^{2}+x - 6)(x + 5)&=x^{2}\cdot x + x^{2}\cdot 5 + x\cdot x + x\cdot 5 - 6\cdot x - 6\cdot 5\\ &=x^{3}+5x^{2}+x^{2}+5x - 6x - 30\\ &=x^{3}+(5x^{2}+x^{2})+(5x - 6x)-30\\ &=x^{3}+6x^{2}-x - 30 \end{align*}$$

\]

To factor \(x^{3}-5x^{2}-4x + 20\), we can use grouping. Group the first two terms and the last two terms:
\[

$$\begin{align*} (x^{3}-5x^{2})+(-4x + 20)&=x^{2}(x - 5)-4(x - 5)\\ &=(x^{2}-4)(x - 5) \end{align*}$$

\]
But \(x^{2}-4\) is a difference of squares, so we can factor it further: \(x^{2}-4=(x + 2)(x - 2)\). So the fully factored form is \((x + 2)(x - 2)(x - 5)\). Let's check the other options:

• Option \((x^{2}-4)(x + 5)\): Expanding this gives \(x^{3}+5x^{2}-4x - 20\), which is not the original polynomial.
• Option \((x^{2}-4)(x - 5)\): This is a partial factorization (since \(x^{2}-4\) can be factored further), but the question asks for linear factors, so we need to factor \(x^{2}-4\) into \((x + 2)(x - 2)\).
• Option \((x + 2)(x - 2)(x + 5)\): Expanding this gives \(x^{3}+5x^{2}-4x - 20\), which is not the original polynomial.

Answer:

\(x^3+6x^2-x-30\)

Question 8