Special Products
E. PRODUCT OF A BINOMIAL AND A TRINOMIAL To find the product of a binomial and a trinomial
1. Cube the first term of the binomial.
2. Add the cube of the last term of the binomial.
In symbols, (a + b)(a2 – ab + b2) = a3 + b3 (a – b)(a2 + ab + b2) = a3 – b3
Examples: 1. (4x2 – 6xy + 9y2)(2x + 3y) = (2x)3 + (3y)3= 8x3 + 27y3 2. (a2 + 2ab + 4b2)(a – 2b) = (a)3 – (2b)3= a3 – 8b3
Mathematics 7 / Second Quarter 51
Perform the indicated operations.
1. (2x + 7)(x – 8) 2. (5a – b2)(3a + 4b2)
3. (16 + 23 ) (14 + 12 )
4. (3 + 2y)(3 – 2y)
5. (7a2 + 11b2)(7a2 – 11b2)
6. (25 − 3) (25 + 3)
7. (–0.8 + z2)2 8. (4c4 – d3)2
9.(34 2 +15 )2
10. (x2y – z)3 11. (–3m + 5n3)3
12. (13 − 14 )3
13. (c – 7)(c2 + 7c + 49) 14. (3a + b)(9a2 – 3ab + b2)
15. (4x2 – 5y)(16x4 + 20x2y + 25y2)
52 Mathematics 7 / Second Quarter
Measurement is the act or process of quantifying anything by comparing it with a known standard unit. The most widely used system of measurement is the SI System which is the modern form of the metric system. The base units for the SI system are the meter, kilogram, second, kelvin, ampere, and mole.
An algebraic expression is a group of terms separated by the plus or minus sign. A constant is a number or a term without a variable while a variable is a symbol, usually a letter that represents an unknown number.
Evaluating an algebraic expression means to substitute the given specific values for the variables and then simplify the resulting numerical expression.
Translating a verbal phrase into an algebraic expression, is to assign a variable to one unknown quantity. Then, write an expression for any other unknown quantities involved in terms of that variable.
A polynomial is a kind of algebraic expression that represents a sum of one or more terms containing whole number exponents on the variables. It can be a monomial, binomial, trinomial, or a multinomial. A polynomial is in standard form if its terms are arranged from the term with the highest degree, up to the term with the lowest degree.
In an, a is called the base and n is called the exponent. The exponent tells the number of times the base is to be used as a factor. Therefore, an = a x a x a…(n times).
The following Laws of Exponents are useful in simplifying algebraic expressions:
Law # 1:Product of Powers with the Same Base For all x, xm ∙ xn = xm+n.
Law # 2:Quotient of Powers with the Same Base For allx, = xm—n.
Law # 3: Raising Power to a Power For all x, (xm)n = xmn. Law # 4: Power of a Product
For all x and y, (xy)m = xmym. Law # 5: Power of a Quotient
For all x and y, ( ) = .
Mathematics 7 / Second Quarter 53
Law # 6: Zero Exponents For all x, x0 =
Law # 7: Negative Exponents For all x, x-n = and − = xn.
To add polynomials, simply combine similar terms. To combine similar terms, get the sum of the numerical coefficients and annex the same literal coefficients.If there are more than one term, for convenience, write similar terms in the same column.
To subtract polynomials, change the sign of the subtrahend then proceed to the addition rule. Also, remember what subtraction means. It is adding the negative of the quantity.
To multiply a monomial by another monomial, simply multiply the numerical coefficients then multiply the literal coefficients by applying the basic laws of exponents.
To multiply a monomial by a polynomial, simply apply the distributive property and follow the rule in multiplying monomial by a monomial.
To multiply a binomial by another binomial, simply distribute the first term of the first binomial to each term of the second binomial then distribute the second term to each term of the other binomial and simplify the results by combining similar terms. This procedure is also known as FOIL method.
To multiply a polynomial with more than one term by a polynomial with three or more terms, simply distribute the first term of the first polynomial to each term of the other polynomial. Repeat the procedure up to the last term and simplify the results by combining similar terms.
In dividing a polynomial by a monomial, divide each term of the polynomial by the monomial. Always apply the laws of exponents to divide out common factors that occur in the dividend and divisor. Then always simplify the resulting fraction.
In dividing a polynomial by a polynomial with more than one term (by long division), there are prescribed steps to follow. To check if the quotient is correct, use the formula (quotient x divisor) + remainder.
Products of polynomials that follow particular patterns are called special products.These are useful shortcuts in multiplying polynomials. Special products could also be of great help in dividing polynomials.
54 Mathematics 7 / Second Quarter
I. Encircle the letter of the correct answer.
1. 0.075 cm2 = _______m2.
a. 0.000075m2 b. 0.0000075 m2 c. 0.00000075 m2 d. 0.000000075 m2 2. Adrianne weighs 120 lbs. How much does he weigh in kilograms?
a. 48.54 kg b. 54.55 kg c. 60.00 kg d. 64.50 kg 3. Luis can run at 12 km/hr. What distance can he cover in 45 minutes?
a. 3 km b. 5 km c. 7 km d. 9 km
4. The numerical coefficient of 12x3y4 is
a. 3 b. 4 c. 12 d. x3y4
5. Which of the following algebraic expressions is a polynomial?
a. (x2 – y) ÷ (x2 + y) b. x + 3 5 – 4
2 1 2
c. 3 + √ – x – 5x d. √3 x – √8 + 2.7x
6.
b2 contain?
How many terms does the expression 10a – √2 a2 – 3ab + √3
a. 5 b. 4 c. 3 d. 2
7. The degree of the polynomial a3b4 + 7a6 – 6b5 + 11 is
a. 5 b. 6 c. 7 d. 18
8. The verbal phrase “eight subtracted from thrice a number” when expressed as algebraic expression is
a. 8 – 3x b. 3x – 8 c. 8 – 2x d. 2x – 8 9. (x-2y-3)0 is equivalent to
a. x2y3 b.
1
c. 0 d. 1
2 3
2−3
10. −7 equals
2
2
-10 b.2102
-4 24a. c. d.
11. The expression (2x-3)-2 written with positive exponent is equal to
a. 1 b. 6 c. 4 d. 5
4 6 4 6 4
12. (6mk)(5m6) =
a. 30m6k b. 30mk/6 c. 30mk+6 d. 30mk-6
Mathematics 7 / Second Quarter 55
13. The expression (x + y)3 is equal to
a. x3 + y3 b. (x + y)(x2 – xy + y2) c. (x + y)(x2+xy + y2) d. x3 + 3x2y + 3xy2 + y3
14. The product of (3x + 5y) and (3x – 5y) is
a. 9x2 – 25y2 b. 9x2 + 30xy – 25y2 c. 9x2 + 15xy – 25y2 d. 9x2 + 25y2
15. What will be the result when x and y are replaced by 3 and 2, respectively, in the expression (2x2y–2)(5x–3y)?
a. 10 b. 5 c. 2 d. 3
3 3 3 5
II. Solve/answer the following completely.
16. The sum of two numbers is 15. Write a polynomial that represents the product of the two numbers.
17. Subtract –m2 + n2 – mn from 0.
18. What should be added to a2 + ab + b2 to obtain 2a2 + 3ab?
19. Subtract 3m – n – mn from the sum of 3m – n + 2mn and – n – mn.
20. Multiply x2 + 2xy + y2 by x – y.
21. Find the product of 25x2 + 37x – 4 and 3x2 + x + 2.
22. Find the width of a rectangular mirror with length of (x + 6) inches and surface area of (x2 + 2x – 24) square inches.
23. Divide 9a3 + 12a2 + 12a – 5 by 3a + 5.
24. Find the quotient: –15x5y6 ÷ 3x2y4.
25. Mico was asked to simplify 2 + 2 −15. His solution is presented below.
2−9
2 + 2 −15
=
( − 3)( + 5)
=
( + 5)
=
5
2− 9 ( − 3)( + 3) ( + 3) 3
Was the solution of Mico correct? Justify your answer.
56 Mathematics 7 / Second Quarter
In a water refilling station, pipe A can fill a tank in 45 minutes, pipe B can fill the same tank in three fifths of an hour, while pipe C can fill it in 2x hours.
a. How many tanks are filled by pipe A in 6 hours? in 12 hours? in a day?
b. How many tanks can pipe C fill in 10 hours ? in4x + 12 hours? in8x2 + 140x hours?
c. Which of the two pipes A and B is more efficient? Explain your answer and
Your work will be graded according to the following criteria.
RUBRICS
CRITERIA Excellent Proficient Adequate Inadequate
4 3 2 1
Mathematical Explanation Explanation Explanation Explanation Reasoning shows thorough shows shows gaps in shows illogical
reasoning and substantial reasoning. reasoning.
justifications. reasoning.
Accuracy of All computations All computations Most of the Some of the the are correct and are correct. computations computations Computations shown in detail. are correct. are correct.
Mathematics 7 / Second Quarter 57