Express $(x-1)(x+2)(x-3)$ in standard form.

To express ( x − 1 ) ( x + 2 ) ( x − 3 ) in standard form, we first expand and then simplify it to get x 3 − 2 x 2 − 5 x + 6 . The final result represents the polynomial in its standard form. This polynomial is fully expanded and simplified from its original factored version.
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Answered by Anonymous | 2025-07-08

Multiply the first two factors: ( x − 1 ) ( x + 2 ) = x 2 + x − 2 .
Multiply the result by the third factor: ( x 2 + x − 2 ) ( x − 3 ) = x 3 − 3 x 2 + x 2 − 3 x − 2 x + 6 .
Simplify the expression: x 3 − 2 x 2 − 5 x + 6 .
The polynomial in standard form is: x 3 − 2 x 2 − 5 x + 6 ​ .

Explanation

Understanding the Problem We are given the polynomial in factored form: ( x − 1 ) ( x + 2 ) ( x − 3 ) . Our goal is to express this polynomial in standard form, which means expanding the product and writing it in the form a x 3 + b x 2 + c x + d .

Multiplying the First Two Factors First, let's multiply the first two factors: ( x − 1 ) ( x + 2 ) .


( x − 1 ) ( x + 2 ) = x ( x + 2 ) − 1 ( x + 2 ) = x 2 + 2 x − x − 2 = x 2 + x − 2 .

Multiplying by the Third Factor Now, let's multiply the result by the third factor: ( x 2 + x − 2 ) ( x − 3 ) .

( x 2 + x − 2 ) ( x − 3 ) = x 2 ( x − 3 ) + x ( x − 3 ) − 2 ( x − 3 ) = x 3 − 3 x 2 + x 2 − 3 x − 2 x + 6 = x 3 − 2 x 2 − 5 x + 6 .

Final Result So, the polynomial in standard form is x 3 − 2 x 2 − 5 x + 6 .

Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, or the behavior of electrical circuits. In this case, expanding the factored form of a polynomial into standard form allows us to easily identify the coefficients and understand the polynomial's behavior. For example, if we want to find the y-intercept of the polynomial, we can simply look at the constant term in the standard form, which is 6 in this case. This means the graph of the polynomial intersects the y-axis at the point (0, 6).

Answered by GinnyAnswer | 2025-07-08