Question

name ren benny date 12/10/25 period multiplying polynomials notes n12 polynomial - an expression with one or more ____ that are joined by addition, subtraction, and multiplication example: 12y² + 5y - 3 monomial - a polynomial with __ term example: 6x³ binomial - a polynomial with __ terms example: x² + 7 distributive property - multiplying a term by ____ terms example: 3x²(2x⁴ - 5) multiplying monomials by polynomials 1. 2x(6x⁴ + x³) 2. y(7y⁵ + 8y²) 3. (8x³)(4x⁵ - xy + 2x) 4. -9x³(-2x⁵ + y - 4x) you try: 1. -3x⁴(8x⁴ - 2x⁷) 2. 5y²(4x³y + x²y - 2xy³) © lindsay bowden, 2

Explanation:

Step1: Recall Distributive Property

The distributive property states that \(a(b + c)=ab + ac\). For the problem \(-3x^{4}(8x^{4}-2x^{7})\), we apply this property by multiplying \(-3x^{4}\) with each term inside the parentheses.

Step2: Multiply the First Terms

Multiply \(-3x^{4}\) and \(8x^{4}\). Using the rule of exponents \(a^{m}\cdot a^{n}=a^{m + n}\), we have \(-3x^{4}\cdot8x^{4}=(-3\times8)x^{4 + 4}=-24x^{8}\).

Step3: Multiply the Second Terms

Multiply \(-3x^{4}\) and \(-2x^{7}\). Again, using the exponent rule: \(-3x^{4}\cdot(-2x^{7})=(-3\times(-2))x^{4+7}=6x^{11}\).

Step4: Combine the Results

Combine the two products from Step 2 and Step 3. So, \(-3x^{4}(8x^{4}-2x^{7})=-24x^{8}+6x^{11}\).

For the second problem \(5y^{2}(4x^{3}y + x^{2}y-2xy^{3})\):

Step1: Apply Distributive Property

Using \(a(b + c + d)=ab+ac + ad\), multiply \(5y^{2}\) with each term inside the parentheses.

Step2: Multiply with First Term

Multiply \(5y^{2}\) and \(4x^{3}y\). Using exponent rule for \(y\) (\(y^{m}\cdot y^{n}=y^{m + n}\)): \(5y^{2}\cdot4x^{3}y=(5\times4)x^{3}y^{2 + 1}=20x^{3}y^{3}\).

Step3: Multiply with Second Term

Multiply \(5y^{2}\) and \(x^{2}y\): \(5y^{2}\cdot x^{2}y = 5x^{2}y^{2+1}=5x^{2}y^{3}\).

Step4: Multiply with Third Term

Multiply \(5y^{2}\) and \(-2xy^{3}\): \(5y^{2}\cdot(-2xy^{3})=(5\times(-2))x y^{2 + 3}=-10xy^{5}\).

Step5: Combine the Results

Combine the products from Step 2, Step 3, and Step 4. So, \(5y^{2}(4x^{3}y + x^{2}y-2xy^{3})=20x^{3}y^{3}+5x^{2}y^{3}-10xy^{5}\).

Answer:

For \(-3x^{4}(8x^{4}-2x^{7})\): \(\boldsymbol{-24x^{8}+6x^{11}}\)

For \(5y^{2}(4x^{3}y + x^{2}y-2xy^{3})\): \(\boldsymbol{20x^{3}y^{3}+5x^{2}y^{3}-10xy^{5}}\)