Math

Question Expand the expression (3x2)(x+5)(3 x-2)(x+5) and find the coefficients of the resulting quadratic expression.

Studdy Solution

STEP 1

Assumptions
1. We are given a binomial expression (3x2)(x+5)(3x - 2)(x + 5).
2. We need to expand this expression to find the coefficients of x2x^2 and xx.

STEP 2

To expand the given expression, we will use the distributive property (also known as the FOIL method for binomials), which states that (a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd.

STEP 3

Identify the terms in the given expression that will be multiplied together.
1. The first terms are 3x3x and xx.
2. The outer terms are 3x3x and 55.
3. The inner terms are 2-2 and xx.
4. The last terms are 2-2 and 55.

STEP 4

Multiply the first terms together to find the coefficient of x2x^2.
Firstterms=3x×x=3x2First\, terms = 3x \times x = 3x^2

STEP 5

Multiply the outer terms together to find part of the coefficient of xx.
Outerterms=3x×5=15xOuter\, terms = 3x \times 5 = 15x

STEP 6

Multiply the inner terms together to find another part of the coefficient of xx.
Innerterms=2×x=2xInner\, terms = -2 \times x = -2x

STEP 7

Multiply the last terms together. This will give us the constant term, but since the problem does not require it, we do not need to calculate it.

STEP 8

Combine the like terms from the outer and inner products to find the total coefficient of xx.
Coefficientofx=15x2x=13xCoefficient\, of\, x = 15x - 2x = 13x

STEP 9

Now we have the expanded form of the given expression.
(3x2)(x+5)=3x2+13x+(constant term)(3x - 2)(x + 5) = 3x^2 + 13x + \text{(constant term)}

STEP 10

The coefficient of x2x^2 is 3 and the coefficient of xx is 13.
Therefore, the expanded expression is 3x2+13x3x^2 + 13x - \square, where the square at the end represents the constant term that we did not calculate.

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