Which of the following represents the completely factored form of the quadratic expression 3x2−3x−18?
3(x−3)(x+2)
3(x+3)(x−2)
(3x−9)(x+2)
3(x−6)(x+1)
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Cevap: 3(x−3)(x+2)
Cevap
The completely factored form is 3(x−3)(x+2).
The correct answer properly extracts the GCF of 3 first, leaving the trinomial x2−x−6. This trinomial factors into (x−3)(x+2) because −3 and 2 multiply to −6 and add up to −1.
Adım Adım Çözüm
1
Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression.
3(x2−x−6)
The coefficients 3,−3, and −18 are all divisible by 3.
2
Find two numbers that multiply to the constant term (−6) and add to the coefficient of the middle term (−1).
The numbers are −3 and 2.
−3×2=−6 and −3+2=−1.
3
Rewrite the quadratic trinomial as the product of two binomials using these numbers.
3(x−3)(x+2)
This represents the final, completely factored form of the original expression.
Anahtar Kavram
Factoring out a Greatest Common Factor (GCF) followed by factoring a quadratic trinomial.
Pair the given polynomial expressions on the left with their fully factored equivalents on the right.
Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın
Öğeler
4x2−25
x2−8x+15
3x2+12x−15
2x3−8x
Eşleşmeler
Cevabı ve açıklamayı göster
Cevap
The correct pairings are: 4x2−25 with (2x−5)(2x+5); x2−8x+15 with (x−3)(x−5); 3x2+12x−15 with 3(x+5)(x−1); and 2x3−8x with 2x(x−2)(x+2).
Each polynomial expression was broken down into its fundamental algebraic components by identifying patterns such as the difference of squares, standard trinomial factoring, and extracting the greatest common factor.
Adım Adım Çözüm
1
Analyze the first expression 4x2−25.
It factors to (2x−5)(2x+5).
It follows the difference of squares pattern a2−b2=(a−b)(a+b) with a=2x and b=5.
2
Analyze the second expression x2−8x+15.
It factors to (x−3)(x−5).
The constant term is positive and the middle coefficient is negative, so both factors must contain negative constants that multiply to 15 and add to −8.
3
Analyze the third expression 3x2+12x−15.
It factors to 3(x+5)(x−1).
Factoring out the GCF of 3 gives 3(x2+4x−5). The resulting quadratic factors into binomials with constants 5 and −1.
4
Analyze the fourth expression 2x3−8x.
It factors to 2x(x−2)(x+2).
Factoring out the GCF of 2x gives 2x(x2−4). The expression x2−4 is a difference of squares that further factors into (x−2)(x+2).