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Sagot :
Alright, Mark, let’s carefully pair each expression in its factored and standard form step by step.
1. Expanding [tex]\((x+1)(x+5)\)[/tex]:
First, we'll use the distributive property (FOIL method) to expand this product of binomials:
[tex]\[ (x+1)(x+5) = x(x+5) + 1(x+5) \][/tex]
Simplifying each term:
[tex]\[ = x^2 + 5x + x + 5 \][/tex]
Combining like terms:
[tex]\[ = x^2 + 6x + 5 \][/tex]
2. Expanding [tex]\((x+2)(x+3)\)[/tex]:
Again, using the distributive property:
[tex]\[ (x+2)(x+3) = x(x+3) + 2(x+3) \][/tex]
Simplifying each term:
[tex]\[ = x^2 + 3x + 2x + 6 \][/tex]
Combining like terms:
[tex]\[ = x^2 + 5x + 6 \][/tex]
3. Reviewing the terms:
Now compare the expanded forms with the given standard form expressions:
[tex]\[ \text{Standard form: } x^2 + 6x + 5 \][/tex]
This corresponds to:
[tex]\[ \text{Factored form: } (x+1)(x+5) \][/tex]
[tex]\[ \text{Standard form: } x^2 + 5x + 6 \][/tex]
This corresponds to:
[tex]\[ \text{Factored form: } (x+2)(x+3) \][/tex]
Given the standard form expression:
[tex]\[ x^2 + 7x + 6 \][/tex]
4. Matching Factored Form to Standard Form:
Comparing each expanded version with [tex]\(x^2 + 7x + 6\)[/tex]:
- [tex]\(x^2 + 6x + 5\)[/tex] does not match with [tex]\(x^2 + 7x + 6\)[/tex].
- [tex]\(x^2 + 5x + 6\)[/tex] does not match with [tex]\(x^2 + 7x + 6\)[/tex].
Thus, we need to factor [tex]\(x^2 + 7x + 6\)[/tex]:
[tex]\[ \text{Factoring: } x^2 + 7x + 6 = (x+1)(x+6) \][/tex]
5. Completing Matches:
Thus, Mark should match:
- [tex]\((x+1)(x+5)\)[/tex] with [tex]\(x^2 + 6x + 5\)[/tex]
- [tex]\((x+2)(x+3)\)[/tex] with [tex]\(x^2 + 5x + 6\)[/tex]
- [tex]\((x+1)(x+6)\)[/tex] with [tex]\(x^2 + 7x + 6\)[/tex]
By following these detailed steps, we identify the correct pairings of factored and standard forms for each expression Mark is working on.
Summary of matches:
- [tex]\(x^2 + 6x + 5\)[/tex] matches [tex]\((x+1)(x+5)\)[/tex]
- [tex]\(x^2 + 5x + 6\)[/tex] matches [tex]\((x+2)(x+3)\)[/tex]
- [tex]\(x^2 + 7x + 6\)[/tex] matches [tex]\((x+1)(x+6)\)[/tex]
1. Expanding [tex]\((x+1)(x+5)\)[/tex]:
First, we'll use the distributive property (FOIL method) to expand this product of binomials:
[tex]\[ (x+1)(x+5) = x(x+5) + 1(x+5) \][/tex]
Simplifying each term:
[tex]\[ = x^2 + 5x + x + 5 \][/tex]
Combining like terms:
[tex]\[ = x^2 + 6x + 5 \][/tex]
2. Expanding [tex]\((x+2)(x+3)\)[/tex]:
Again, using the distributive property:
[tex]\[ (x+2)(x+3) = x(x+3) + 2(x+3) \][/tex]
Simplifying each term:
[tex]\[ = x^2 + 3x + 2x + 6 \][/tex]
Combining like terms:
[tex]\[ = x^2 + 5x + 6 \][/tex]
3. Reviewing the terms:
Now compare the expanded forms with the given standard form expressions:
[tex]\[ \text{Standard form: } x^2 + 6x + 5 \][/tex]
This corresponds to:
[tex]\[ \text{Factored form: } (x+1)(x+5) \][/tex]
[tex]\[ \text{Standard form: } x^2 + 5x + 6 \][/tex]
This corresponds to:
[tex]\[ \text{Factored form: } (x+2)(x+3) \][/tex]
Given the standard form expression:
[tex]\[ x^2 + 7x + 6 \][/tex]
4. Matching Factored Form to Standard Form:
Comparing each expanded version with [tex]\(x^2 + 7x + 6\)[/tex]:
- [tex]\(x^2 + 6x + 5\)[/tex] does not match with [tex]\(x^2 + 7x + 6\)[/tex].
- [tex]\(x^2 + 5x + 6\)[/tex] does not match with [tex]\(x^2 + 7x + 6\)[/tex].
Thus, we need to factor [tex]\(x^2 + 7x + 6\)[/tex]:
[tex]\[ \text{Factoring: } x^2 + 7x + 6 = (x+1)(x+6) \][/tex]
5. Completing Matches:
Thus, Mark should match:
- [tex]\((x+1)(x+5)\)[/tex] with [tex]\(x^2 + 6x + 5\)[/tex]
- [tex]\((x+2)(x+3)\)[/tex] with [tex]\(x^2 + 5x + 6\)[/tex]
- [tex]\((x+1)(x+6)\)[/tex] with [tex]\(x^2 + 7x + 6\)[/tex]
By following these detailed steps, we identify the correct pairings of factored and standard forms for each expression Mark is working on.
Summary of matches:
- [tex]\(x^2 + 6x + 5\)[/tex] matches [tex]\((x+1)(x+5)\)[/tex]
- [tex]\(x^2 + 5x + 6\)[/tex] matches [tex]\((x+2)(x+3)\)[/tex]
- [tex]\(x^2 + 7x + 6\)[/tex] matches [tex]\((x+1)(x+6)\)[/tex]
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