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Question 281Question →

Which of the following represents the fully simplified product of the polynomials (3a−2b)(3a - 2b) and (4a2+5ab−b2)(4a^2 + 5ab - b^2)?

Show answer & explanation

Answer: 12a3+7a2b−13ab2+2b312a^3 + 7a^2b - 13ab^2 + 2b^3

Answer

The fully simplified product is 12a3+7a2b−13ab2+2b312a^3 + 7a^2b - 13ab^2 + 2b^3.
The correct answer is obtained by distributing each term in the binomial to every term in the trinomial, and then combining the like terms. Multiplying 3a3a by each term yields 12a3+15a2b−3ab212a^3 + 15a^2b - 3ab^2. Multiplying −2b-2b by each term yields −8a2b−10ab2+2b3-8a^2b - 10ab^2 + 2b^3. Adding these together and combining the like terms (15a2b−8a2b=7a2b15a^2b - 8a^2b = 7a^2b and −3ab2−10ab2=−13ab2-3ab^2 - 10ab^2 = -13ab^2) results in the final expression.

Step-by-Step Solution

1
Distribute the first term of the binomial, 3a3a, to every term in the trinomial.
3a(4a2)+3a(5ab)+3a(−b2)=12a3+15a2b−3ab23a(4a^2) + 3a(5ab) + 3a(-b^2) = 12a^3 + 15a^2b - 3ab^2
This applies the distributive property to begin multiplying the polynomials.
2
Distribute the second term of the binomial, −2b-2b, to every term in the trinomial.
−2b(4a2)−2b(5ab)−2b(−b2)=−8a2b−10ab2+2b3-2b(4a^2) - 2b(5ab) - 2b(-b^2) = -8a^2b - 10ab^2 + 2b^3
This completes the distribution process for all terms.
3
Write out the full expanded expression by combining the results from the previous steps.
12a3+15a2b−3ab2−8a2b−10ab2+2b312a^3 + 15a^2b - 3ab^2 - 8a^2b - 10ab^2 + 2b^3
This provides the complete polynomial before simplification.
4
Group and combine the like terms.
12a3+(15a2b−8a2b)+(−3ab2−10ab2)+2b3=12a3+7a2b−13ab2+2b312a^3 + (15a^2b - 8a^2b) + (-3ab^2 - 10ab^2) + 2b^3 = 12a^3 + 7a^2b - 13ab^2 + 2b^3
Combining terms with identical variable parts simplifies the expression to its final standard form.

Key Concept

Multiplying a binomial by a polynomial requires distributing every term in the first polynomial to every term in the second polynomial, carefully tracking negative signs, and then combining any like terms.
Estimated Time:1m 30s
Question 282Question →

Calculate the value of ww in the given linear equation:

Fill in the blanks below

3(2w+5)−4w=5(w−1)+2 3(2w + 5) - 4w = 5(w - 1) + 2

Value of ww:
Show answer & explanation

Answer

The correct value for the variable is 6.
The correct answer is 6. Applying the distributive property to both sides simplifies the equation to 6w+15−4w=5w−5+26w + 15 - 4w = 5w - 5 + 2. Grouping like terms on each side yields 2w+15=5w−32w + 15 = 5w - 3. Subtracting 2w2w from both sides and adding 33 to both sides results in 18=3w18 = 3w. Dividing both sides by 3 isolates the variable, giving w=6w = 6.

Step-by-Step Solution

1
Distribute the constants across the terms inside the parentheses on both sides.
6w+15−4w=5w−5+26w + 15 - 4w = 5w - 5 + 2
To eliminate the parentheses and expose all terms for simplification.
2
Combine the like terms on each side of the equation independently.
2w+15=5w−32w + 15 = 5w - 3
To simplify both expressions before manipulating terms across the equals sign.
3
Move all variable terms to one side of the equation.
Subtract 2w2w from both sides to obtain 15=3w−315 = 3w - 3.
To group all the terms containing ww together and avoid negative coefficients.
4
Isolate the variable term and solve for ww.
Add 3 to both sides to get 18=3w18 = 3w, then divide by 3 to find w=6w = 6.
To determine the final numerical value of the variable.

Key Concept

Solving multi-step linear equations utilizing the distributive property and combining like terms.
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