To factor the expression \( 2x^4 + 19x^3 + 35x^2 \), we can start by factoring out the greatest common factor (GCF) from the expression.
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Identify the GCF of the expression: The GCF of the terms \( 2x^4, 19x^3, \) and \( 35x^2 \) is \( x^2 \).
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Factor out \( x^2 \): \[ 2x^4 + 19x^3 + 35x^2 = x^2(2x^2 + 19x + 35) \]
Next, we need to factor the quadratic \( 2x^2 + 19x + 35 \).
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To factor \( 2x^2 + 19x + 35 \), we can look for two numbers that multiply to \( 2 \times 35 = 70 \) and add to \( 19 \). The numbers that fit are \( 14 \) and \( 5 \).
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Rewrite \( 19x \): \[ 2x^2 + 14x + 5x + 35 \]
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Group the terms: \[ (2x^2 + 14x) + (5x + 35) \]
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Factor each group: \[ 2x(x + 7) + 5(x + 7) \]
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Factor out the common binomial factor \( (x + 7) \): \[ (2x + 5)(x + 7) \]
Now, we have fully factored the expression: \[ 2x^4 + 19x^3 + 35x^2 = x^2(2x + 5)(x + 7) \]
Thus, the linear and quadratic factors are:
- Linear factors: \( x + 7 \), \( 2x + 5 \)
- Quadratic factor: \( x^2 \) (note that while this is technically a quadratic, it doesn't have linear terms).
So, in conclusion:
- The factors are \( x^2 \), \( (2x + 5) \), and \( (x + 7) \).
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