What is 12×3 – 9×2 – 4x 3 in factored form? ( x2 – )( x – ).
The Correct Answer and Explanation is :
To simplify and factor the expression ( 12x^3 – 9x^2 – 4x ), we will first factor out the greatest common factor (GCF) from all the terms. Here’s the step-by-step breakdown:
- Find the GCF:
The terms are ( 12x^3 ), ( -9x^2 ), and ( -4x ).
The GCF of these terms is ( x ), since all terms have at least one factor of ( x ). - Factor out the GCF:
Factoring out ( x ) from the expression:
[
12x^3 – 9x^2 – 4x = x(12x^2 – 9x – 4)
] - Factor the quadratic:
Now, we need to factor the quadratic part ( 12x^2 – 9x – 4 ).
To do this, we use factoring by grouping:
- Multiply the leading coefficient (12) by the constant (-4): ( 12 \times -4 = -48 ).
- Find two numbers that multiply to give -48 and add to give the middle term (-9). These numbers are -12 and +4, because:
[
-12 \times 4 = -48 \quad \text{and} \quad -12 + 4 = -9
]
- Rewrite the middle term using -12 and 4:
[
12x^2 – 9x – 4 = 12x^2 – 12x + 4x – 4
] - Factor by grouping:
Group the terms in pairs:
[
(12x^2 – 12x) + (4x – 4)
]
Factor each group:
[
12x(x – 1) + 4(x – 1)
]
Now factor out the common binomial factor ( (x – 1) ):
[
(12x + 4)(x – 1)
]
- Simplify further:
Factor out the GCF from the first binomial:
[
4(3x + 1)(x – 1)
]
So, the factored form of the original expression is:
[
x \cdot 4(3x + 1)(x – 1)
]
Thus, the complete factored form of ( 12x^3 – 9x^2 – 4x ) is:
[
4x(3x + 1)(x – 1)
]
Explanation:
Factoring is the process of breaking down an expression into simpler terms that, when multiplied together, give the original expression. In this case, we first identified that ( x ) was a common factor among all terms and factored it out. The next step involved factoring the remaining quadratic expression. We used the technique of factoring by grouping, which required finding two numbers that both multiply and add in a way that matches the coefficients in the quadratic. After factoring by grouping, we simplified the binomial further by factoring out common terms. This led us to the final factored form.
Factoring helps in simplifying expressions and solving equations, especially in algebra, making it a crucial skill for working with polynomials.