Math / Algebra

QuestionFactor. $s^{2}+8 s d+15 d^{2}$

Studdy Solution

STEP 1

1. The expression is a quadratic trinomial in terms of $s$ and $d$ .
2. We are looking for two binomials that multiply to give the original expression.

STEP 2

1. Identify the structure of the quadratic trinomial.
2. Determine the factors of the quadratic trinomial.

STEP 3

Identify the structure of the quadratic trinomial. The expression is in the form $ax^2 + bx + c$ , where: - $a = 1$ (coefficient of $s^2$ ), - $b = 8d$ (coefficient of $sd$ ), - $c = 15d^2$ (constant term).

STEP 4

Determine the factors of the quadratic trinomial. We need to find two numbers that multiply to $a \cdot c = 1 \cdot 15d^2 = 15d^2$ and add to $b = 8d$ .
The numbers that satisfy these conditions are $3d$ and $5d$ .

STEP 5

Rewrite the middle term $8sd$ using the numbers found:
$s^2 + 3sd + 5sd + 15d^2$

STEP 6

Factor by grouping:
Group the terms: $(s^2 + 3sd) + (5sd + 15d^2)$ .
Factor each group:
$s(s + 3d) + 5d(s + 3d)$

STEP 7

Factor out the common binomial factor:
$(s + 3d)(s + 5d)$
The factored form of the expression is:
$\boxed{(s + 3d)(s + 5d)}$

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QuestionFactor. $s^{2}+8 s d+15 d^{2}$

Studdy Solution

STEP 1

1. The expression is a quadratic trinomial in terms of $s$ and $d$ .
2. We are looking for two binomials that multiply to give the original expression.

STEP 2

1. Identify the structure of the quadratic trinomial.
2. Determine the factors of the quadratic trinomial.

STEP 3

Identify the structure of the quadratic trinomial. The expression is in the form $ax^2 + bx + c$ , where: - $a = 1$ (coefficient of $s^2$ ), - $b = 8d$ (coefficient of $sd$ ), - $c = 15d^2$ (constant term).

STEP 4

Determine the factors of the quadratic trinomial. We need to find two numbers that multiply to $a \cdot c = 1 \cdot 15d^2 = 15d^2$ and add to $b = 8d$ .
The numbers that satisfy these conditions are $3d$ and $5d$ .

STEP 5

Rewrite the middle term $8sd$ using the numbers found:
$s^2 + 3sd + 5sd + 15d^2$

STEP 6

Factor by grouping:
Group the terms: $(s^2 + 3sd) + (5sd + 15d^2)$ .
Factor each group:
$s(s + 3d) + 5d(s + 3d)$

STEP 7

Factor out the common binomial factor:
$(s + 3d)(s + 5d)$
The factored form of the expression is:
$\boxed{(s + 3d)(s + 5d)}$

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QuestionFactor. s2+8sd+15d2s^{2}+8 s d+15 d^{2}s2+8sd+15d2

Studdy Solution

STEP 1

1. The expression is a quadratic trinomial in terms of sss and ddd.2. We are looking for two binomials that multiply to give the original expression.

STEP 2

1. Identify the structure of the quadratic trinomial.2. Determine the factors of the quadratic trinomial.

STEP 3

Identify the structure of the quadratic trinomial. The expression is in the form ax2+bx+c ax^2 + bx + c ax2+bx+c, where: - a=1 a = 1 a=1 (coefficient of s2 s^2 s2), - b=8d b = 8d b=8d (coefficient of sd sd sd), - c=15d2 c = 15d^2 c=15d2 (constant term).

STEP 4

Determine the factors of the quadratic trinomial. We need to find two numbers that multiply to a⋅c=1⋅15d2=15d2 a \cdot c = 1 \cdot 15d^2 = 15d^2 a⋅c=1⋅15d2=15d2 and add to b=8d b = 8d b=8d.The numbers that satisfy these conditions are 3d 3d 3d and 5d 5d 5d.

STEP 5

Rewrite the middle term 8sd 8sd 8sd using the numbers found:s2+3sd+5sd+15d2 s^2 + 3sd + 5sd + 15d^2 s2+3sd+5sd+15d2

STEP 6

Factor by grouping:Group the terms: (s2+3sd)+(5sd+15d2) (s^2 + 3sd) + (5sd + 15d^2) (s2+3sd)+(5sd+15d2).Factor each group:s(s+3d)+5d(s+3d) s(s + 3d) + 5d(s + 3d) s(s+3d)+5d(s+3d)

STEP 7

Factor out the common binomial factor:(s+3d)(s+5d) (s + 3d)(s + 5d) (s+3d)(s+5d) The factored form of the expression is:(s+3d)(s+5d) \boxed{(s + 3d)(s + 5d)} (s+3d)(s+5d)​

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QuestionFactor. $s^{2}+8 s d+15 d^{2}$

1. The expression is a quadratic trinomial in terms of $s$ and $d$ .
2. We are looking for two binomials that multiply to give the original expression.

1. Identify the structure of the quadratic trinomial.
2. Determine the factors of the quadratic trinomial.

Identify the structure of the quadratic trinomial. The expression is in the form $ax^2 + bx + c$ , where: - $a = 1$ (coefficient of $s^2$ ), - $b = 8d$ (coefficient of $sd$ ), - $c = 15d^2$ (constant term).

Determine the factors of the quadratic trinomial. We need to find two numbers that multiply to $a \cdot c = 1 \cdot 15d^2 = 15d^2$ and add to $b = 8d$ .
The numbers that satisfy these conditions are $3d$ and $5d$ .

Rewrite the middle term $8sd$ using the numbers found:
$s^2 + 3sd + 5sd + 15d^2$

Factor by grouping:
Group the terms: $(s^2 + 3sd) + (5sd + 15d^2)$ .
Factor each group:
$s(s + 3d) + 5d(s + 3d)$

Factor out the common binomial factor:
$(s + 3d)(s + 5d)$
The factored form of the expression is:
$\boxed{(s + 3d)(s + 5d)}$