To factor the polynomial \( x^2 + 10x + 21 \) using the identity \( x^2 + (a + b)x + ab = (x + a)(x + b) \), follow these steps:
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Identify the coefficients: From the polynomial \( x^2 + 10x + 21 \), we can identify:
- The coefficient of \( x \) (which is \( a + b \)) is \( 10 \).
- The constant term (which is \( ab \)) is \( 21 \).
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Set up the equations: We need to find two numbers, \( a \) and \( b \), such that:
- \( a + b = 10 \)
- \( ab = 21 \)
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Find suitable \( a \) and \( b \): We can look for pairs of factors of \( 21 \) that add up to \( 10 \). The pairs of factors of \( 21 \) are:
- \( (1, 21) \)
- \( (3, 7) \)
Now we check which pair adds up to \( 10 \):
- \( 1 + 21 = 22 \) (not suitable)
- \( 3 + 7 = 10 \) (suitable)
Therefore, we have \( a = 3 \) and \( b = 7 \).
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Use the identity: Using the values of \( a \) and \( b \), we can now express the polynomial in its factored form: \[ x^2 + 10x + 21 = (x + a)(x + b) = (x + 3)(x + 7). \]
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Final factored form: The factored form of the polynomial \( x^2 + 10x + 21 \) is: \[ (x + 3)(x + 7). \]
Thus, the factored form of the polynomial is \( (x + 3)(x + 7) \).
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