Answer :
Sure! Let's determine the other factor of the polynomial \( x^2 + 8x + 15 \) knowing that \( x + 5 \) is one of its factors.
1. Identify the given polynomial:
We are given the polynomial \( x^2 + 8x + 15 \).
2. Identify the given factor:
We are given that \( x + 5 \) is a factor of the polynomial.
3. Express the polynomial in terms of its factors:
Since \( x + 5 \) is a factor, the polynomial \( x^2 + 8x + 15 \) can be written as:
[tex]\[ x^2 + 8x + 15 = (x + 5)(\text{other factor}) \][/tex]
4. Determine the other factor:
Let's denote the other factor by \( (x + b) \).
5. Equate the expression to match the polynomial:
Now, matching the polynomial, we expand:
[tex]\[ (x + 5)(x + b) = x^2 + (5 + b)x + 5b \][/tex]
By matching coefficients, we have:
[tex]\[ x^2 + 8x + 15 = x^2 + (5 + b)x + 5b \][/tex]
6. Compare coefficients of \( x \):
For the coefficient of \( x \):
[tex]\[ 8 = 5 + b \implies b = 3 \][/tex]
7. Compare the constant terms:
For the constant term:
[tex]\[ 15 = 5b \implies 15 = 5 \cdot 3 \implies b = 3 \][/tex]
8. Identify the other factor:
Therefore, the other factor is \( x + 3 \).
So, the other factor of the polynomial \( x^2 + 8x + 15 \) given that \( x + 5 \) is one of its factors is:
[tex]\[ \boxed{x + 3} \][/tex]
1. Identify the given polynomial:
We are given the polynomial \( x^2 + 8x + 15 \).
2. Identify the given factor:
We are given that \( x + 5 \) is a factor of the polynomial.
3. Express the polynomial in terms of its factors:
Since \( x + 5 \) is a factor, the polynomial \( x^2 + 8x + 15 \) can be written as:
[tex]\[ x^2 + 8x + 15 = (x + 5)(\text{other factor}) \][/tex]
4. Determine the other factor:
Let's denote the other factor by \( (x + b) \).
5. Equate the expression to match the polynomial:
Now, matching the polynomial, we expand:
[tex]\[ (x + 5)(x + b) = x^2 + (5 + b)x + 5b \][/tex]
By matching coefficients, we have:
[tex]\[ x^2 + 8x + 15 = x^2 + (5 + b)x + 5b \][/tex]
6. Compare coefficients of \( x \):
For the coefficient of \( x \):
[tex]\[ 8 = 5 + b \implies b = 3 \][/tex]
7. Compare the constant terms:
For the constant term:
[tex]\[ 15 = 5b \implies 15 = 5 \cdot 3 \implies b = 3 \][/tex]
8. Identify the other factor:
Therefore, the other factor is \( x + 3 \).
So, the other factor of the polynomial \( x^2 + 8x + 15 \) given that \( x + 5 \) is one of its factors is:
[tex]\[ \boxed{x + 3} \][/tex]
