determine the $y$-intercept of the function $f(x) = x^4 + 3x^3 - 2x + 7$.

a) $(0, 7)$
b) $(0, -7)$
c) $(0, 9)$
d) $(0, 5)$

question 2 (5 points)
what are the real solutions to the equation $4x^3 - x^2 - 4x + 1 = 0$?

a) $x = -1, 1/4, 1$
b) $x = -1, 1$
c) $x = -1, 0, 1/4$
d) $x = -1, -1/4, 1$

question 3 (5 points)
use synthetic division to find the quotient if polynomial $p(x) = x^3 - 7x + 6$ is divided by $x - 2$.

a) $x^2 - 2x - 3$
b) $x^2 - 2x + 3$
c) $x^2 + 2x + 3$
d) $x^2 + 2x - 3$

Question

determine the $y$-intercept of the function $f(x) = x^4 + 3x^3 - 2x + 7$.

a) $(0, 7)$
b) $(0, -7)$
c) $(0, 9)$
d) $(0, 5)$

question 2 (5 points)
what are the real solutions to the equation $4x^3 - x^2 - 4x + 1 = 0$?

a) $x = -1, 1/4, 1$
b) $x = -1, 1$
c) $x = -1, 0, 1/4$
d) $x = -1, -1/4, 1$

question 3 (5 points)
use synthetic division to find the quotient if polynomial $p(x) = x^3 - 7x + 6$ is divided by $x - 2$.

a) $x^2 - 2x - 3$
b) $x^2 - 2x + 3$
c) $x^2 + 2x + 3$
d) $x^2 + 2x - 3$

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "y-intercept of polynomial", "Factoring by Grouping", "Synthetic Division" ], "current_concepts": [ "y-intercept of polynomial", "Factoring by Grouping", "Synthetic Division" ] } ## Evaluate the function at x = 0 $$ \begin{aligned} f(0) &= 0^4 + 3(0)^3 - 2(0) + 7 \ f(0) &= 7 \end{aligned} $$ ## Factor the cubic equation by grouping $$ \begin{aligned} 4x^3 - x^2 - 4x + 1 &= 0 \ x^2(4x - 1) - 1(4x - 1) &= 0 \ (x^2 - 1)(4x - 1) &= 0 \ (x - 1)(x + 1)(4x - 1) &= 0 \ x &= 1, -1, \frac{1}{4} \end{aligned} $$ ## Perform synthetic division $$ \begin{array}{c|rrrr} 2 & 1 & 0 & -7 & 6 \ & & 2 & 4 & -6 \ \hline & 1 & 2 & -3 & 0 \end{array} $$ $$ \text{Quotient} = x^2 + 2x - 3 $$ ## Question 1 A) (0, 7)

B) (0, -7)

C) (0, 9)

D) (0, 5) ## Question 2 A) $x = -1, 1/4, 1$

B) $x = -1, 1$

C) $x = -1, 0, 1/4$

D) $x = -1, -1/4, 1$ ## Question 3 A) $x^2 - 2x - 3$

B) $x^2 - 2x + 3$

C) $x^2 + 2x + 3$

D) $x^2 + 2x - 3$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Division and Roots" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate the function at x = 0

Factor the cubic equation by grouping

Perform synthetic division

Answer:

Question 1

Question 2

Question 3