56. Multiplication and division of polynomials. In sections 20 and 21 the pupil has seen that,

a(b+c+d)=ab+ac+ad,
(a+b)(c+d)=ac+ad+bc+bd,

and has learned the

Rule. To multiply two polynomials together multiply each term of the multiplicand by each term of the multiplier and add these products.

This can now be applied when some of the terms of the polynomials are negative.

EXAMPLE. Multiply 2x-3 by 3x+4.

In section 23 the pupil learned to divide a polynomial by a monomial by dividing each term of the polynomial by the monomial. He can now apply this rule to negative numbers. EXAMPLE. Divide -4s2+5rs-3s by -8.

SOLUTION. (-4s2+5rs-3s)÷( − s) = +4s-5r +3, found by dividing -4s2 by −s, then dividing +5rs by −s, then −3s by

-S.

9. (m-n) (m+n) = ?
10. (3x-7y) (3x+7y) = ?
11. (x2+4)(x2-4)=?
12. (a-b) (a-b) = ?
13. (3a-b)(3a—b) = ?
14. (x-y)2=?

15. (−xy+2y) ÷ (− y) = ?
16. (3x-12)(-3)=?

17. (15ab-12ac-9ax)÷(+3a) = ?

Exercise 62. Review of signed numbers

1. Make a number scale and show the following numbers: +8; +3; 6; 9; 14; -8; +8-5; -8+5; -12-2; +4-6; +3+2-5; -7+9-3+5; -1-1+2 +2-2.

2. Find the average of the following thermometer readings: +10°, +4°, −9°, −12°, −5°, 0°, −1o, +3°, and +7°. 3. The average monthly temperature of a place in Russia for a certain year was for January, 42°; for February, -29°; for March, -8°; for April, +4°; for May, +18°; for June, +43°; for July, +58°; for August, +61°; for September, +38°; for October, +6°; for November, -8°; for December, -26°. Find the average annual temperature.

4. What is meant by the absolute value of a number? Read the absolute value of -6; of +18; of the sum of -15 and +3; of the difference when +9 is subtracted from -4.

5. Give the sum, the difference, the product, and the quotient of 12 and +3; of. +6a and -2a.

6.

(−6)(+7)+(+4)(−6) — ?
(-7)-(+4)

=

7. If given the date of a man's birth and the date of his death, how can you find his age at the time cf his death? Use this plan to find the age at death of Augustus Cæsar who was born in the year -63 and died in the year +14.

8. How old at death was Cicero who was born in the year 106 B.C. and died in the year 43 B.C.?

9. Socrates was born 469 B.C. Confucius died 478 B.C. How long before the birth of Socrates did Confucius die?

10. What can be added to -2x to get -7x? To +3a to get -5a?

11. By what can -x be multiplied to get +x? By what -x be divided to get +x?

can

12. What can be added to x-3 to get x? What can be added to x+3 to get x?

13. By what can be multiplied to get x?

14. What can be added to 2x-3 to get -3? To -7 to get 0?

15. 3-1-4-7=?

16. -a-a- ?

19. x-2(x)=?

20. x(x)=?

17. (-a)(-a)+2a(-a)=? 21. −32÷(−3)2 = ?

18. 22-(-2)2 = ?

22. If a=-4 and b=2, find the value of b-2a2b.

23. If r=-1 and s=-2, find the value of r2-2rs; of s2+rs+r2.

CHAPTER VII

EQUATIONS AND PROBLEMS

57. Transposing terms in an equation.

The pupil has already learned how to solve simple equations and problems not involving negative numbers. He will find that many of the processes are made much simpler by using the knowledge of negative numbers which he now has.

What number added to +4 gives 0 as a result?

What number added to −2x gives 0 as a result? What number added to 2x-3 gives 2x as a result? What number added to 2x+3 gives 2x as a result? What number added to each member of the equation 3x-a-b gives 3x for the left member?

Add n to each member of the equation

Observe that any term is eliminated, that is, removed from a member of the equation, by adding to both members that term with its sign changed.

Observe, too, that the term eliminated from one member appears in the other member with its sign changed.

Rule. Any term may be changed from one member of an equation to the other provided its sign is changed.

This process is called transposition and the term is said to be transposed from one member of the equation to the other.

EXAMPLE 1. Solve the equation 3x-7=5x-13.

SOLUTION.

Transposing 5 and -7,

Combining terms,

Dividing by -2,
Checking,

3x-7=5x-13. 3x-5x=-13+7.

-2x=-6.

x = 3.
3.3-7=5.3—13.

2=2.

Steps in solving equations like 2x+5=7x-4.

(1) Transpose the unknown terms to one member and the known terms to the other.

(2) Combine like terms.

(3) Divide each member by the coefficient of the unknown. (4) Check by substituting in the original equation the value found for the unknown number.

EXAMPLE 2. Solve for x the equation, 5x+a=7a+3x.

15. Solve for a, a-bc=d. Solve also for b; for c.

16. Solve for n, mn-m2=0.

17. Solve for x, (x−2)h=3h.

18. Solve for h, (x-2)h=3h+hx-5.