Question

question 20 of 25
according to the synthetic division below, which of the following statements
are true?
check all that apply.
\\( 5\overline{|4\quad -17\quad -15} \\)
\\( \quad\quad\quad\quad 20\quad\quad 15 \\)
\\( \quad\overline{\quad 4\quad\quad 3\quad\quad\\ 0} \\)
\\( \square \\) a. \\( (4x^2 - 17x - 15) \div (x - 5) = (4x + 3) \\)
\\( \square \\) b. \\( (x - 5) \\) is a factor of \\( 4x^2 - 17x - 15 \\).
\\( \square \\) c. \\( (4x^2 - 17x - 15) \div (x + 5) = (4x + 3) \\)
\\( \square \\) d. the number \\( -5 \\) is a root of \\( f(x) = 4x^2 - 17x - 15 \\).
\\( \square \\) e. the number \\( 5 \\) is a root of \\( f(x) = 4x^2 - 17x - 15 \\).
\\( \square \\) f. \\( (x + 5) \\) is a factor of \\( 4x^2 - 17x - 15 \\).

Explanation:

Step1: Identify dividend and divisor

Dividend: $4x^2 -17x -15$; Divisor root: 5 → divisor $(x-5)$

Step2: Check remainder

Remainder = 0 → division exact

Step3: Identify quotient

Quotient: $4x + 3$

Step4: Evaluate statements

A: $(4x^2-17x-15)÷(x-5)=4x+3$ ✔️
B: $(x-5)$ is factor (remainder 0) ✔️
C: Divisor is $(x+5)$? No ❌
D: Root is -5? No (root is 5) ❌
E: 5 is root (F(5)=0) ✔️
F: $(x+5)$ is factor? No ❌

Answer:

A. $(4x^{2}-17x-15)\div(x-5)=(4x+3)$, B. $(x - 5)$ is a factor of $4x^{2}-17x-15$, E. The number 5 is a root of $F(x)=4x^{2}-17x-15$