Factor polynomials in the form 'a² + 2ab + b²'.
Understand the Problem
The question is asking how to factor polynomials in the form of 'a² + 2ab + b²', which is known as the identity of perfect squares. It provides several examples with step-by-step solutions to factor the given expressions.
Answer
1. $(x + 2)^2$ 2. $(x + 6)^2$ 3. $(x + 1)^2$ 4. $(x + 5)^2$ 5. $(x + 9)^2$ 6. $(4 + y)^2$ 7. $(6 + y)^2$ 8. $(7 + y)^2$
Answer for screen readers
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$ ( x + 2 )^2 $
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$ ( x + 6 )^2 $
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$ x^2 + 2x + 1 = (x + 1)^2 $
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$ x^2 + 10x + 25 = (x + 5)^2 $
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$ x^2 + 18x + 81 = (x + 9)^2 $
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$ 16 + 8y + y^2 = (4 + y)^2 $
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$ 36 + 12y + y^2 = (6 + y)^2 $
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$ 49 + 14y + y^2 = (7 + y)^2 $
Steps to Solve
- Identify the structure of the polynomial
We start by recognizing that the polynomial has the form $a^2 + 2ab + b^2$.
- Match the components of the polynomial
For the polynomial $x^2 + 4x + 4$, we notice:
- $a^2 = x^2$ (so $a = x$)
- $2ab = 4x$ (so $2ab$ suggests $2(x)(b) = 4x$; hence $b = 2$)
- $b^2 = 4$ (leads to $b = 2$)
- Formulate the perfect square
Using the identity $a^2 + 2ab + b^2 = (a + b)^2$, we apply: $$ x^2 + 4x + 4 = (x + 2)^2 $$
- Repeat the process for the next polynomial
For the second polynomial $36 + 12x + x^2$, rearranging gives $x^2 + 12x + 36$:
- $a^2 = x^2$ (so $a = x$)
- $2ab = 12x$ (thus $b = 6$)
- $b^2 = 36$ (reconfirming $b = 6$)
- Formulate the perfect square
Applying the identity again: $$ x^2 + 12x + 36 = (x + 6)^2 $$
- Complete the remaining problems
The same strategy is applied to the other expressions, ensuring to identify $a$ and $b$ properly for each polynomial.
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$ ( x + 2 )^2 $
-
$ ( x + 6 )^2 $
-
$ x^2 + 2x + 1 = (x + 1)^2 $
-
$ x^2 + 10x + 25 = (x + 5)^2 $
-
$ x^2 + 18x + 81 = (x + 9)^2 $
-
$ 16 + 8y + y^2 = (4 + y)^2 $
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$ 36 + 12y + y^2 = (6 + y)^2 $
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$ 49 + 14y + y^2 = (7 + y)^2 $
More Information
These expressions showcase the factoring of perfect square trinomials, which are very useful in algebra for simplifying equations and solving quadratic expressions.
Tips
- Forgetting to rearrange the polynomial in the standard form $a^2 + 2ab + b^2$ before identifying $a$ and $b$.
- Misidentifying $b$, leading to incorrect factors. Always verify by checking that $b^2$ matches the constant term.
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