Adding and Subtracting Polynomials

CHAPTER 5. POLYNOMIALS
298
We’re asked to find the time it takes the projectile to return to ground level.
When this happens, its height y above ground level will equal zero. Substitute
0 for y in the last equation.
0 = 52 + 276t − 4.9t2 .
Load the right-hand side of this equation into the Y= menu of your graphing
calculator, then set the WINDOW parameters as follows: Xmin=0, Xmax=60,
Ymin=0, and Ymax=4000. Push the GRAPH button to produce the graph,
then use the 2:zero tool from the CALC menu to find where the graph of
f crosses the x-axis. Report the results on your homework as shown in the
following graph.
y(meters)
4,000
0
y = 52 + 276t − 4.9t2
0
56.5
60
t(seconds)
Hence, the projectile reaches ground level at approximately 56.5 seconds.
5.4
Adding and Subtracting Polynomials
1. Let’s arrange our answer in descending powers of r by listing the highest
powers of r first, then the next highest, and so on. Use the commutative and
associative properties to change the order and regroup. Then combine like
terms.
(−8r2 t + 7rt2 + 3t3 ) + (9r3 + 2rt2 + 4t3 )
= (9r3 ) + (−8r2 t) + (7rt2 + 2rt2 ) + (3t3 + 4t3 )
= 9r3 − 8r2 t + 9rt2 + 7t3
3. Let’s arrange our answer in descending powers of x by listing the highest
powers of x first, then the next highest, and so on. Use the commutative and
associative properties to change the order and regroup. Then combine like
terms.
(7x2 − 6x − 9) + (8x2 + 10x + 9)
= (7x2 + 8x2 ) + (−6x + 10x) + (−9 + 9)
= 15x2 + 4x
Second Edition: 2012-2013
5.4. ADDING AND SUBTRACTING POLYNOMIALS
299
5. Let’s arrange our answer in descending powers of r by listing the highest
powers of r first, then the next highest, and so on. Use the commutative and
associative properties to change the order and regroup. Then combine like
terms.
(−2r2 + 7rs + 4s2 ) + (−9r2 + 7rs − 2s2 )
= (−2r2 − 9r2 ) + (7rs + 7rs) + (4s2 − 2s2 )
= −11r2 + 14rs + 2s2
7. Let’s arrange our answer in descending powers of y by listing the highest
powers of y first, then the next highest, and so on. Use the commutative and
associative properties to change the order and regroup. Then combine like
terms.
(−8y 3 − 3y 2 z − 6z 3 ) + (−3y 3 + 7y 2 z − 9yz 2 )
= (−8y 3 − 3y 3 ) + (−3y 2 z + 7y 2 z) + (−9yz 2) + (−6z 3 )
= −11y 3 + 4y 2 z − 9yz 2 − 6z 3
9. Negating a polynomial is accomplished by reversing the sign of each term
of the polynomial. Thus,
−(5x2 − 4) = −5x2 + 4
11. Negating a polynomial is accomplished by reversing the sign of each term
of the polynomial. Thus,
−(9r3 − 4r2 t − 3rt2 + 4t3 ) = −9r3 + 4r2 t + 3rt2 − 4t3
13. Negating a polynomial is accomplished by reversing the sign of each term
of the polynomial. Thus,
−(−5x2 + 9xy + 6y 2 ) = 5x2 − 9xy − 6y 2
15. To arrange the answer in descending powers of u, place the highest power
of u first, then the next highest, and so on. First, distribute the minus sign,
changing the sign of each term of the second polynomial.
(−u3 − 4u2 w + 7w3 ) − (u2 w + uw2 + 3w3 )
= −u3 − 4u2 w + 7w3 − u2 w − uw2 − 3w3
Now use the commutative and associative properties to change the order and
regroup. Combine like terms.
= (−u3 ) + (−4u2 w − u2 w) + (−uw2 ) + (7w3 − 3w3 )
= −u3 − 5u2 w − uw2 + 4w3
Second Edition: 2012-2013
CHAPTER 5. POLYNOMIALS
300
17. To arrange the answer in descending powers of y, place the highest power
of y first, then the next highest, and so on. First, distribute the minus sign,
changing the sign of each term of the second polynomial.
(2y 3 − 2y 2 z + 3z 3 ) − (−8y 3 + 5yz 2 − 3z 3 )
= 2y 3 − 2y 2 z + 3z 3 + 8y 3 − 5yz 2 + 3z 3
Now use the commutative and associative properties to change the order and
regroup. Combine like terms.
= (2y 3 + 8y 3 ) + (−2y 2 z) + (−5yz 2) + (3z 3 + 3z 3 )
= 10y 3 − 2y 2 z − 5yz 2 + 6z 3
19. To arrange the answer in descending powers of r, place the highest power
of r first, then the next highest, and so on. First, distribute the minus sign,
changing the sign of each term of the second polynomial.
(−7r2 − 9rs − 2s2 ) − (−8r2 − 7rs + 9s2 )
= −7r2 − 9rs − 2s2 + 8r2 + 7rs − 9s2
Now use the commutative and associative properties to change the order and
regroup. Combine like terms.
= (−7r2 + 8r2 ) + (−9rs + 7rs) + (−2s2 − 9s2 )
= r2 − 2rs − 11s2
21. To arrange the answer in descending powers of x, place the highest power
of x first, then the next highest, and so on. First, distribute the minus sign,
changing the sign of each term of the second polynomial.
(10x2 + 2x − 6) − (−8x2 + 14x + 17)
= 10x2 + 2x − 6 + 8x2 − 14x − 17
Now use the commutative and associative properties to change the order and
regroup. Combine like terms.
= (10x2 + 8x2 ) + (2x − 14x) + (−6 − 17)
= 18x2 − 12x − 23
Second Edition: 2012-2013
5.4. ADDING AND SUBTRACTING POLYNOMIALS
301
23. To arrange our answer in descending powers of x, we need to first list
the highest power of x, then the next highest power, and so on. Use the
commutative and associative properties to change the order and regroup. Then
combine like terms.
f (x) + g(x) = (−2x2 + 9x + 7) + (8x3 − 7x2 + 5)
= (8x3 ) + (−2x2 − 7x2 ) + (9x) + (7 + 5)
= 8x3 − 9x2 + 9x + 12
25. To arrange our answer in descending powers of x, we need to first list
the highest power of x, then the next highest power, and so on. Use the
commutative and associative properties to change the order and regroup. Then
combine like terms.
f (x) + g(x) = (5x3 − 5x2 + 8x) + (7x2 − 2x − 9)
= (5x3 ) + (−5x2 + 7x2 ) + (8x − 2x) + (−9)
= 5x3 + 2x2 + 6x − 9
27. To arrange our answer in descending powers of x, we need to first list
the highest power of x, then the next highest power, and so on. Use the
commutative and associative properties to change the order and regroup. Then
combine like terms.
f (x) + g(x) = (−3x2 − 8x − 9) + (5x2 − 4x + 4)
= (−3x2 + 5x2 ) + (−8x − 4x) + (−9 + 4)
= 2x2 − 12x − 5
29. To arrange our answer in descending powers of x, we need to first list
the highest power of x, then the next highest power, and so on. Use the
commutative and associative properties to change the order and regroup. Then
combine like terms.
f (x) − g(x) = (−6x3 − 7x + 7) − (−3x3 − 3x2 − 8x)
= −6x3 − 7x + 7 + 3x3 + 3x2 + 8x
= (−6x3 + 3x3 ) + (3x2 ) + (−7x + 8x) + (7)
= −3x3 + 3x2 + x + 7
Second Edition: 2012-2013
CHAPTER 5. POLYNOMIALS
302
31. To arrange the answer in descending powers of x, place the highest power
of x first, then the next highest, and so on. First, distribute the minus sign,
changing the sign of each term of the second polynomial.
f (x) − g(x) = (12x2 − 5x + 4) − (8x2 − 16x − 7)
= 12x2 − 5x + 4 − 8x2 + 16x + 7
Now use the commutative and associative properties to change the order and
regroup. Combine like terms.
= (12x2 − 8x2 ) + (−5x + 16x) + (4 + 7)
= 4x2 + 11x + 11
33. To arrange our answer in descending powers of x, we need to first list
the highest power of x, then the next highest power, and so on. Use the
commutative and associative properties to change the order and regroup. Then
combine like terms.
f (x) − g(x) = (−3x3 − 4x + 2) − (−4x3 − 7x2 + 6)
= −3x3 − 4x + 2 + 4x3 + 7x2 − 6
= (−3x3 + 4x3 ) + (7x2 ) + (−4x) + (2 − 6)
= x3 + 7x2 − 4x − 4
35. The two shaded squares in have areas A1 = x2 and A3 = 25, respectively.
The two unshaded rectangles have areas A2 = 5x and A4 = 5x.
x
5
5
A2 = 5x
A3 = 25 5
x
A1 = x2
A4 = 5x x
x
5
Summing these four areas gives us the area of the entire figure.
A = A1 + A2 + A3 + A4
= x2 + 5x + 25 + 5x
= x2 + 10x + 25
Second Edition: 2012-2013