select all the correct answers.

use synthetic division to completely factor this polynomial.

\3x^4 - 14x^3 - 3x^2 + 54x - 40\

which expressions are factors of the polynomial?

- $(3x + 5)$
- $(x - 2)$
- $(x - 4)$
- $(x + 2)$
- $(x - 1)$
- $(x + 4)$
- $(3x - 5)$
- $(x + 1)$

Question

select all the correct answers.

use synthetic division to completely factor this polynomial.

\3x^4 - 14x^3 - 3x^2 + 54x - 40\

which expressions are factors of the polynomial?

- $(3x + 5)$
- $(x - 2)$
- $(x - 4)$
- $(x + 2)$
- $(x - 1)$
- $(x + 4)$
- $(3x - 5)$
- $(x + 1)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Synthetic Division", "Polynomial Factoring" ], "new_concepts": [], "current_concepts": [ "Synthetic Division", "Polynomial Factoring", "Factor Theorem" ] } ## Evaluate potential roots using the Factor Theorem $$ \begin{aligned} P(x) &= 3x^4 - 14x^3 - 3x^2 + 54x - 40 \ P(1) &= 3(1)^4 - 14(1)^3 - 3(1)^2 + 54(1) - 40 = 3 - 14 - 3 + 54 - 40 = 0 \ P(2) &= 3(2)^4 - 14(2)^3 - 3(2)^2 + 54(2) - 40 = 48 - 112 - 12 + 108 - 40 = -8 eq 0 \ P(4) &= 3(4)^4 - 14(4)^3 - 3(4)^2 + 54(4) - 40 = 768 - 896 - 48 + 216 - 40 = 0 \ P(-4) &= 3(-4)^4 - 14(-4)^3 - 3(-4)^2 + 54(-4) - 40 = 768 + 896 - 48 - 216 - 40 = 1360 eq 0 \ P(-2) &= 3(-2)^4 - 14(-2)^3 - 3(-2)^2 + 54(-2) - 40 = 48 + 112 - 12 - 108 - 40 = 0 \end{aligned} $$ ## Perform synthetic division with known roots $$ \begin{array}{r|rrrrr} 1 & 3 & -14 & -3 & 54 & -40 \ & & 3 & -11 & -14 & 40 \ \hline & 3 & -11 & -14 & 40 & 0 \end{array} $$ $$ \begin{array}{r|rrrr} 4 & 3 & -11 & -14 & 40 \ & & 12 & 4 & -40 \ \hline & 3 & 1 & -10 & 0 \end{array} $$ $$ \begin{array}{r|rrr} -2 & 3 & 1 & -10 \ & & -6 & 10 \ \hline & 3 & -5 & 0 \end{array} $$ ## Factor the remaining linear quotient $$ \begin{aligned} P(x) &= (x - 1)(x - 4)(x + 2)(3x - 5) \end{aligned} $$

$(3x + 5)$

$(x - 2)$ $(x - 4)$

$(x + 2)$

$(x - 1)$

$(x + 4)$

$(3x - 5)$

$(x + 1)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Factoring" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate potential roots using the Factor Theorem

Perform synthetic division with known roots

Factor the remaining linear quotient

Answer: