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

Question image

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 \( 9x^3 - 3x^2 - 27x - 14 \).
Answer for screen readers

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

Steps to Solve

  1. 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 ]

  1. 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 ]

  1. 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 ]

  1. 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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