QuestionFactor the expressions:
1. $2 y^{3}+14 y^{2}+24 y$
2. $40 x^{2}-101 x+63 = (A x-B)(C x-D)$ with $A<C$ , $B<D$ .

Studdy Solution

STEP 1

Assumptions1. The expressions given are polynomials. . The polynomials are to be factored completely.
3. The coefficients and powers of the variables in the polynomials are real numbers.

STEP 2

Let's start with the first expression. We can factor out the greatest common factor from each term.
$2y^{} +14y^{2} +24y =2y(y^{2} +7y +12)$

STEP 3

Now, we can factor the quadratic expression inside the parentheses.
$y^{2} +7y +12 = (y +)(y +3)$

STEP 4

Substitute this back into the expression to get the completely factored form of the first expression.
$2y^{3} +14y^{2} +24y =2y(y +4)(y +3)$

STEP 5

Now, let's move on to the second expression. We can write the expression in the form $(Ax - B)(Cx - D)$ .
$40x^{2} -101x +63 = (Ax - B)(Cx - D)$

STEP 6

We know that the product of $A$ and $C$ should give the coefficient of $x^{2}$ , and the product of $B$ and should give the constant term. Also, $A$ and or $B$ and $C$ should give the coefficient of $x$ when multiplied and subtracted.

STEP 7

By trial and error, we can find that $A =5$ , $B =7$ , $C =$ , and $=9$ satisfy these conditions.

STEP 8

Substitute these values back into the expression to get the completely factored form of the second expression.
$40x^{2} -101x +63 = (5x -7)(8x -)$ So, the completely factored forms of the given expressions are $2y(y +4)(y +3)$ and $(5x -7)(8x -)$ .

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