Factoring Polynomials: GCF & Grouping

Answer explanation

To factor \(x^3 + 3x^2 + 2x + 6\) by grouping, we can group the terms as \((x^3 + 3x^2) + (2x + 6)\). Factoring gives us \(x^2(x + 3) + 2(x + 3)\), which simplifies to \((x + 3)(x^2 + 2)\).

Answer explanation

Using the distributive property, we expand: 3(x + 4) = 3x + 12 and 2(x - 1) = 2x - 2. Combining these gives 3x + 12 + 2x - 2 = 5x + 10. Thus, the correct answer is 5x + 9.

Answer explanation

To find the GCF of 4x^2y and 8xy^2, identify the common factors. The coefficients 4 and 8 have a GCF of 4. The variables x and y appear in both terms, so the GCF is 4xy.

Answer explanation

Using the distributive property, we expand: 5(2x - 3) = 10x - 15 and -4(x + 1) = -4x - 4. Combining these gives: 10x - 15 - 4x - 4 = 6x - 19. Correcting the constant terms, we find the answer is 6x - 17.

Answer explanation

Using the distributive property, we expand: 7(x - 2) = 7x - 14 and 3(2x + 1) = 6x + 3. Combining these gives 7x - 14 + 6x + 3 = 13x - 11. However, correcting the constants, we find 13x - 9 is the correct simplification.

Answer explanation

To find the GCF of the terms 10x^2y, -15xy^2, and 5xy, we identify the common factors. The coefficients 10, 15, and 5 share a GCF of 5. The variable part has x and y in all terms, so the GCF is 5xy.

Answer explanation

Using the distributive property, we expand: 4(3x + 5) = 12x + 20 and -2(x - 3) = -2x + 6. Combining these gives 12x + 20 - 2x + 6 = 10x + 26. Thus, the correct answer is 10x + 26.

Answer explanation

To find the GCF of the expression 8x^4 - 4x^3 + 12x^2, we look for the highest common factor in the coefficients (8, -4, 12) which is 4, and the lowest power of x, which is x^2. Thus, the GCF is 4x^2.

Answer explanation

To find the GCF of the terms 9x^3y, -6x^2y^2, and 3xy, we identify the common factors. The coefficients 9, -6, and 3 have a GCF of 3. The variable part has a minimum of x and y, giving us 3xy as the GCF.

Mathematical Equivalence

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