Concept Byte: Dividing Polynomials Using Algebra Tiles

Use With Lesson 11-3

ACTIVITY

You can use algebra tiles to model polynomial division.

Activity

What is ( x 2 + 4 x + 3 ) ÷ ( x + 3 ) ? open , x squared , plus 4 x plus 3 close divides open x plus 3 close question mark  Use algebra tiles.

• Step 1 Use algebra tiles to model the dividend, x 2 + 4 x + 3 . x squared , plus 4 x plus 3 .

  

• Step 2 Use the x 2 x squared -tile and the 1-tiles to form a figure with length x + 3 , x plus 3 comma  the divisor.

  

• Step 3 Use the remaining tiles to fill in the rectangle.

  

Since ( x + 1 ) ( x + 3 ) = x 2 + 4 x + 3 , open x plus 1 close open x plus 3 close equals , x squared , plus 4 x plus 3 comma  you can write ( x 2 + 4 x + 3 ) ÷ ( x + 3 ) = x + 1 . open , x squared , plus 4 x plus 3 close divides open x plus 3 close equals x plus 1 .

Check Check your result by multiplying x + 1 x plus 1  and x + 3 . x plus 3 .  The product should be the dividend, x 2 + 4 x + 3 . x squared , plus 4 x plus 3 .

( x + 1 ) ( x + 3 ) = ( x ) ( x ) + ( x ) ( 3 ) + ( 1 ) ( x ) + ( 1 ) ( 3 )   = x 2 + 3 x + x + 3   = x 2 + 4 x + 3 ✓ table with 3 rows and 2 columns , row1 column 1 , open , x plus 1 , close . open , x plus 3 , close , column 2 equals , open x close . open x close , plus , open x close . open 3 close , plus , open 1 close . open x close , plus , open 1 close . open 3 close , row2 column 1 , , column 2 equals , x squared , plus 3 x plus x plus 3 , row3 column 1 , , column 2 equals , x squared , plus 4 x plus 3 check mark , end table

Exercises

Use algebra tiles to find each quotient. Check your result.

1. ( x 2 + 6 x + 8 ) ÷ ( x + 4 ) open , x squared , plus 6 x plus 8 close divides open x plus 4 close
2. ( x 2 + 5 x + 6 ) ÷ ( x + 2 ) open , x squared , plus 5 x plus 6 close divides open x plus 2 close
3. ( x 2 + 8 x + 12 ) ÷ ( x + 6 ) open , x squared , plus 8 x plus 12 close divides open x plus 6 close
4. ( x 2 + 8 x + 7 ) ÷ ( x + 1 ) open , x squared , plus 8 x plus 7 close divides open x plus 1 close
5. Reasoning In Exercises 1–4, the divisor is a factor of the dividend. How do you know? Can you use algebra tiles to represent polynomial division when the divisor is not a factor of the dividend? Explain.

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