Distributivity and Algebra | IT

Question 33

6.3 Mind the Mistake, Mend the Mistake

We have expanded and simplified some algebraic expressions below to their simplest forms.

(i) Check each of the simplifications and see if there is a mistake. (ii) If there is a mistake, try to explain what could have gone wrong. (iii) Then write the correct expression.

Solution

We will carefully check each algebraic simplification by applying the rules of distribution, exponents, and combining like terms.

Step 1 — Checking Expression 1

Let us look at the first expression. We need to apply the distributive property.

$-3p(-5p + 2q)$

The given simplification incorrectly distributes $-3p$ as only $-3p$ and then adds $5p$ . We must multiply $-3p$ by each term inside the parentheses.

$= (-3p)(-5p) + (-3p)(2q)$

$= 15p^2 - 6pq$

$\boxed{15p^2 - 6pq}$

Step 2 — Checking Expression 2

Let us check the second expression. We need to distribute the numbers outside the parentheses.

$2(x-1) + 3(x+4)$

The given simplification did not multiply $2$ by $-1$ . It should be $2x - 2$ .

$= 2x - 2 + 3x + 12$

Now, we combine the like terms.

$= (2x + 3x) + (-2 + 12)$

$= 5x + 10$

$\boxed{5x + 10}$

Step 3 — Checking Expression 3

Let us examine the third expression. We need to distribute the $2$ and then combine like terms.

$y + 2(y+2)$

The given simplification incorrectly assumed the expression was $(y+2)^2$ . These are different expressions.

$= y + 2y + 4$

Now, we combine the like terms.

$= 3y + 4$

$\boxed{3y + 4}$

Step 4 — Checking Expression 4

Let us look at the fourth expression. We need to square the binomial $(5m + 6n)$ .

$(5m + 6n)^2$

The given simplification missed the middle term. When we square a binomial $(a+b)^2$ , the result is $a^2 + 2ab + b^2$ .

$= (5m)^2 + 2(5m)(6n) + (6n)^2$

$= 25m^2 + 60mn + 36n^2$

$\boxed{25m^2 + 60mn + 36n^2}$

Step 5 — Checking Expression 5

Let us check the fifth expression. We need to square the binomial $(-q + 2)$ .

$(-q + 2)^2$

The given simplification is correct. We can use the formula $(a+b)^2 = a^2 + 2ab + b^2$ .

$= (-q)^2 + 2(-q)(2) + (2)^2$

$= q^2 - 4q + 4$

$\boxed{q^2 - 4q + 4}$

Step 6 — Checking Expression 6

Let us examine the sixth expression. We need to multiply the terms.

$3a(2b \times 3c)$

The given simplification incorrectly distributed $3a$ to both $2b$ and $3c$ . We should first multiply the terms inside the parentheses.

$= 3a(6bc)$

$= 18abc$

$\boxed{18abc}$

Step 7 — Checking Expression 7

Let us look at the seventh expression. We need to distribute $\frac{1}{2}$ and then combine constants.

$\frac{1}{2}(10s - 6) + 3$

The given simplification is correct. We distribute $\frac{1}{2}$ to each term inside the parentheses.

$= \frac{1}{2}(10s) - \frac{1}{2}(6) + 3$

$= 5s - 3 + 3$

$= 5s$

$\boxed{5s}$

Step 8 — Checking Expression 8

Let us check the eighth expression. We need to combine the terms.

$5w^2 + 6w$

The given simplification incorrectly added unlike terms. Terms can only be added if they have the same variable raised to the same power. $w^2$ and $w$ are not like terms.

This expression cannot be simplified further.

$\boxed{5w^2 + 6w}$

Step 9 — Checking Expression 9

Let us examine the ninth expression. We need to combine the terms.

$2a^2 + 3a^3 + 6a^2b + 6ab^2$

The given simplification incorrectly added unlike terms. All terms in the original expression have different variable combinations or powers.

This expression cannot be simplified further.

$\boxed{2a^2 + 3a^3 + 6a^2b + 6ab^2}$

Step 10 — Checking Expression 10

Let us look at the tenth expression. We need to multiply the two binomials.

$(x+2)(x+5)$

The given simplification is correct. We use the distributive property (also known as FOIL method) to multiply each term.

$= x(x+5) + 2(x+5)$

$= x^2 + 5x + 2x + 10$

$= x^2 + 7x + 10$

$\boxed{x^2 + 7x + 10}$

Step 11 — Checking Expression 11

Let us check the eleventh expression. We need to multiply the two binomials.

$(a+2)(b+4)$

The given simplification only multiplied the first terms ( $a \cdot b$ ) and the last terms ( $2 \cdot 4$ ). It missed the inner and outer products.

$= a(b+4) + 2(b+4)$

$= ab + 4a + 2b + 8$

$\boxed{ab + 4a + 2b + 8}$

Step 12 — Checking Expression 12

Let us examine the twelfth expression. We need to factor out the common term.

$ab^2 + a^2b + a^2b^2$

The given simplification is correct. We can see that $ab$ is a common factor in all terms.

$= ab \cdot b + ab \cdot a + ab \cdot ab$

$= ab(b + a + ab)$

We can rearrange the terms inside the parentheses.

$= ab(a + b + ab)$

$\boxed{ab(a + b + ab)}$

Answer

(i) Expressions 1, 2, 3, 4, 6, 8, 9, and 11 contain mistakes. (ii) For Expression 1, the distributive property was not applied to all terms inside the parentheses. For Expression 2, the distributive property was not applied correctly to the first term, $2(x-1)$ . For Expression 3, the expression was incorrectly assumed to be a squared binomial, $(y+2)^2$ . For Expression 4, the middle term ( $2ab$ ) was missed when squaring the binomial $(a+b)^2$ . For Expression 6, the multiplication inside the parentheses was incorrectly distributed with the outside term. For Expression 8, unlike terms ( $w^2$ and $w$ ) were incorrectly added together. For Expression 9, unlike terms were incorrectly added and combined. For Expression 11, only the first and last terms were multiplied when expanding the binomials. (iii) The correct expressions are: Expression 1: $\mathbf{15p^2 - 6pq}$ Expression 2: $\mathbf{5x + 10}$ Expression 3: $\mathbf{3y + 4}$ Expression 4: $\mathbf{25m^2 + 60mn + 36n^2}$ Expression 6: $\mathbf{18abc}$ Expression 8: $\mathbf{5w^2 + 6w}$ Expression 9: $\mathbf{2a^2 + 3a^3 + 6a^2b + 6ab^2}$ Expression 11: $\mathbf{ab + 4a + 2b + 8}$

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