Expand the expression to a polynomial in standard form: (3x + 2)(3x² - 3x - 7)

Understand the Problem

The question is asking to expand the expression (3x + 2)(3x² - 3x - 7) into a polynomial and write it in standard form.

Answer

The expanded form is \( 9x^3 - 3x^2 - 27x - 14 \).

Steps to Solve

Distributing the first term of the first binomial

Start by distributing the first term ( 3x ) from the first binomial ( (3x + 2) ) to each term in the second binomial ( (3x^2 - 3x - 7) ).

[ 3x \cdot 3x^2 = 9x^3 ] [ 3x \cdot (-3x) = -9x^2 ] [ 3x \cdot (-7) = -21x ]

Distributing the second term of the first binomial

Now distribute the second term ( 2 ) from the first binomial ( (3x + 2) ) to each term in the second binomial ( (3x^2 - 3x - 7) ).

[ 2 \cdot 3x^2 = 6x^2 ] [ 2 \cdot (-3x) = -6x ] [ 2 \cdot (-7) = -14 ]

Combining like terms

Add all the terms from steps 1 and 2 together and combine like terms.

[ 9x^3 + (-9x^2 + 6x^2) + (-21x - 6x) - 14 ]

This simplifies to:

[ 9x^3 - 3x^2 - 27x - 14 ]

Writing in standard form

Ensure the polynomial is written in standard form, which arranges terms by descending powers of ( x ).

The standard form is:

[ 9x^3 - 3x^2 - 27x - 14 ]

The expanded expression in standard form is ( 9x^3 - 3x^2 - 27x - 14 ).

More Information

The process of expanding a binomial expression involves distributing each term of one binomial across all terms of the other binomial. This technique ensures that each term is accounted for in the final polynomial.

Tips

Neglecting to distribute all terms correctly.
Forgetting to combine like terms properly.
Writing terms in a non-standard order (not arranging by decreasing powers of ( x )).

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