To factor the polynomial \( x^2 + 10x + 21 \) using the identity \( x^2 + (a+b)x + ab = (x+a)(x+b) \), we need to identify the values of \( a \) and \( b \) that satisfy this identity.
Here are the steps to find the factored form:
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Identify \( a + b \) and \( ab \):
- In the polynomial \( x^2 + 10x + 21 \):
- \( a + b = 10 \)
- \( ab = 21 \)
- In the polynomial \( x^2 + 10x + 21 \):
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Find \( a \) and \( b \):
- We need to find two numbers \( a \) and \( b \) that satisfy both conditions: their sum is 10, and their product is 21.
- To do this, we'll list pairs of factors of 21:
- \( (1, 21) \)
- \( (3, 7) \)
- Now, we check which pair sums to 10:
- \( 1 + 21 = 22 \) (not a match)
- \( 3 + 7 = 10 \) (this matches)
Thus, we have found \( a = 3 \) and \( b = 7 \).
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Use the factors to write the polynomial in factored form:
- Now that we have \( a = 3 \) and \( b = 7 \), we can use the identity: \[ x^2 + 10x + 21 = (x + 3)(x + 7) \]
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Final Result: Therefore, the factored form of the polynomial \( x^2 + 10x + 21 \) is: \[ (x + 3)(x + 7) \]
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