practice problems
6. add: $(4x^2 - 3x + 7) + (-2x^2 + 5x - 1)$
7. subtract: $(6x^2 + 2x - 9) - (3x^2 - 4x + 5)$
8. multiply using foil: $(2x + 3)(x - 4)$
9. multiply using a special product formula: $(x + 6)^2$
10. multiply: $(3x - 5)(3x + 5)$

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Accepted Answer

{ "quality": "clear", "question_count": 5, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Polynomial Addition", "Polynomial Subtraction", "Polynomial Multiplication" ], "current_concepts": [ "Polynomial Addition", "Polynomial Subtraction", "Polynomial Multiplication", "FOIL Method", "Special Products" ] } ## Simplify Question 6 Combine like terms: $$ \begin{aligned} (4x^2 - 3x + 7) + (-2x^2 + 5x - 1) &= (4x^2 - 2x^2) + (-3x + 5x) + (7 - 1) \ &= 2x^2 + 2x + 6 \end{aligned} $$ ## Simplify Question 7 Distribute the negative sign and combine like terms: $$ \begin{aligned} (6x^2 + 2x - 9) - (3x^2 - 4x + 5) &= 6x^2 + 2x - 9 - 3x^2 + 4x - 5 \ &= (6x^2 - 3x^2) + (2x + 4x) + (-9 - 5) \ &= 3x^2 + 6x - 14 \end{aligned} $$ ## Simplify Questions 8, 9, and 10 Apply multiplication methods: - For Question 8 (FOIL): $$ \begin{aligned} (2x + 3)(x - 4) &= 2x(x) + 2x(-4) + 3(x) + 3(-4) \ &= 2x^2 - 8x + 3x - 12 \ &= 2x^2 - 5x - 12 \end{aligned} $$ - For Question 9 (Perfect Square Trinomial): $$ \begin{aligned} (x + 6)^2 &= x^2 + 2(x)(6) + 6^2 \ &= x^2 + 12x + 36 \end{aligned} $$ - For Question 10 (Difference of Squares): $$ \begin{aligned} (3x - 5)(3x + 5) &= (3x)^2 - 5^2 \ &= 9x^2 - 25 \end{aligned} $$ | No. | Problem | Answer | |---|---|---| | 6 | Add: $(4x^2 - 3x + 7) + (-2x^2 + 5x - 1)$ | $2x^2 + 2x + 6$ | | 7 | Subtract: $(6x^2 + 2x - 9) - (3x^2 - 4x + 5)$ | $3x^2 + 6x - 14$ | | 8 | Multiply using FOIL: $(2x + 3)(x - 4)$ | $2x^2 - 5x - 12$ | | 9 | Multiply using a special product formula: $(x + 6)^2$ | $x^2 + 12x + 36$ | | 10 | Multiply: $(3x - 5)(3x + 5)$ | $9x^2 - 25$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Operations" ] }

No.	Problem	Answer
7	Subtract: \((6x^2 + 2x - 9) - (3x^2 - 4x + 5)\)	\(3x^2 + 6x - 14\)
8	Multiply using FOIL: \((2x + 3)(x - 4)\)	\(2x^2 - 5x - 12\)
9	Multiply using a special product formula: \((x + 6)^2\)	\(x^2 + 12x + 36\)
10	Multiply: \((3x - 5)(3x + 5)\)	\(9x^2 - 25\)

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