The expression for the area of these rectangles is x(10 − x) or 10x − x2.
1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. Can you point out any similarity or difference between the algebraic expressions obtained for Raju’s pens and pencils (4x + 5y + 3) and for these rectangles?
- 1. Can you identify the terms, variables and coefficients of this algebraic expression?
- 2. Can you point out any similarity or difference between the algebraic expressions obtained for Raju’s pens and pencils (4x + 5y + 3) and for these rectangles?
Answer: 1. Terms: 10x and −x2; variable: x; coefficients: 10 (of x) and −1 (of x2). 2. Both are algebraic expressions with terms and coefficients, but 4x + 5y + 3 uses two variables, each only to power 1, and has a constant; 10x − x2 uses one variable, has an x2 term and no constant.
Step-by-step solution
Idea: Write the expression as a sum of terms first: 10x − x2 = 10x + (−1)x2. The sign in front of a term belongs to its coefficient.
1. Can you identify the terms, variables and coefficients of this algebraic expression?
- Use the expanded form 10x − x2 = 10x + (−x2). The terms are 10x and −x2.½ mark
- Only one letter appears, so the only variable is x.½ mark
- Coefficients: 10 is the coefficient of x; −x2 means (−1) × x2, so the coefficient of x2 is −1.½ mark
Terms 10x, −x2; variable x; coefficients 10 and −1.
2. Can you point out any similarity or difference between the algebraic expressions obtained for Raju’s pens and pencils (4x + 5y + 3) and for these rectangles?
- Similarity: both are algebraic expressions made of terms with numerical coefficients, and both contain a term in x with a positive coefficient (4x and 10x).½ mark
- Difference in variables: 4x + 5y + 3 has two variables, x and y; 10x − x2 has only one variable, x.½ mark
- Difference in powers and constants: in 4x + 5y + 3 every variable has power 1, and there is a constant 3. In 10x − x2 the variable appears squared (x2), and there is no constant term.½ mark
Both have terms and coefficients; but the first has two variables (all to power 1) and a constant, the second has one variable, an x2 term and no constant.
1. Terms 10x and −x2, variable x, coefficients 10 and −1. 2. Both are expressions with terms and coefficients; 4x + 5y + 3 has two variables of power 1 and a constant, while 10x − x2 has one variable, a squared term and no constant.
Answer to write in the exam
1.
10x − x2 = 10x + (−1)x2
Terms: 10x, −x2
Variable: x
Coefficients: 10 (of x), −1 (of x2)
2.
Similarity: both are algebraic expressions with terms and numerical coefficients.
4x + 5y + 3 has two variables; 10x − x2 has one variable.
4x + 5y + 3 has only power-1 terms and a constant 3; 10x − x2 has an x2 term and no constant.
Common mistakes that cost marks
- Giving the coefficient of x2 as 1 instead of −1. The minus sign in front of x2 is part of its coefficient.
- Listing 10 and x as separate terms. 10x is one term: 10 multiplied by x.
- Counting x and x2 as two different variables. There is one variable, x, appearing with two different powers.
How this can come in the exam
MCQ (1 mark)
The coefficient of t2 in 8t − t2 + 5 is
- 1
- −1
- 8
- 0
Show answer
(B) −1
−t2 = (−1)t2, so the coefficient is −1.
Short answer (2 marks)
Write two differences between the expressions 3m + 2n − 7 and 6m − m2.
Show answer
(1) 3m + 2n − 7 has two variables (m, n); 6m − m2 has one (m) (1 mark). (2) The second has a squared term m2, while the first has only power-1 terms and a constant −7 (1 mark).Try one yourself
Write the terms, the variable and the coefficients of 12y − 3y2.
Show answer
Terms 12y, −3y2; variable y; coefficient of y is 12 and of y2 is −3.