Question

Simplify
(i) \[({x^2} - 5)(x + 5) + 25\]
(ii) $({a^2} + 5)({b^2} + 3) + 5$
(iii) \[(t + {8^2})({t^2} - s)\]
(iv) \[(a + b)(c - d) + (a - b)(c + d) + 2(ac + bd)\]
(v) \[(x + y)(2x + y) + (x + 2y)(x - y)\]
(vi) \[(x + y)({x^2} - xy + {y^2})\]
(vii) \[(1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y\]
(viii) \[(a + b + c)(a + b - c)\]

Answer 1

Hint: Multiply each term of the first bracket with each term of the second bracket to get a polynomial. Combine the like terms if there are any and simplify the polynomial.

Complete step by step solution:
We are given 8 expressions.
 We need to simplify these expressions
(i) Given binomials \[({x^2} - 5)\] and \[(x + 5)\]in the expression \[({x^2} - 5)(x + 5) + 25\].
Consider the product \[({x^2} - 5)(x + 5)\]. We will add 25 to the product later.
We will first multiply each term of the binomial \[({x^2} - 5)\]with each term of the binomial\[(x + 5)\].
Thus, we have \[({x^2} - 5)(x + 5) = {x^2} \times (x + 5) - 5(x + 5)\]
Here we will multiply the term outside a bracket with each of the terms inside that bracket to obtain a polynomial with 4 terms and then we add 25 to it.
This gives us\[({x^2} - 5)(x + 5) = ({x^2} \times x + {x^2} \times 5) - (5 \times x + 5 \times 5) = ({x^3} + 5{x^2}) - (5x + 25)\]
Now\[({x^2} - 5)(x + 5) + 25 = ({x^3} + 5{x^2}) - (5x + 25) + 25\]
Now we will combine the like terms (the underlined part) to simplify the equation and get the final answer.
\[({x^2} - 5)(x + 5) + 25 = {x^3} + 5{x^2} - 5x\underline { - 25 + 25} = {x^3} + 5{x^2} - 5x + 0 = {x^3} + 5{x^2} - 5x\]
Hence the simplified expression is \[{x^3} + 5{x^2} - 5x\].

We will be repeating the above method for the next 7 expressions.
(ii) Consider the expression $({a^2} + 5)({b^2} + 3) + 5$
$
  ({a^2} + 5)({b^2} + 3) + 5 = {a^2}({b^2} + 3) + 5({b^2} + 3) + 5 \\
   = ({a^2} \times {b^2} + {a^2} \times 3) + (5 \times {b^2} + 5 \times 3) + 5 \\
   = ({a^2}{b^2} + 3{a^2}) + (5{b^2} + 15) + 5 \\
   = {a^2}{b^2} + 3{a^2} + 5b\underline { + 15 + 5} \\
   = {a^2}{b^2} + 3{a^2} + 5b + 20 \\
 $
Hence the simplified expression is ${a^2}{b^2} + 3{a^2} + 5b + 20$.
(iii) Consider the expression \[(t + {8^2})({t^2} - s)\]
\[
  (t + {8^2})({t^2} - s) = t({t^2} - s) + {8^2}({t^2} - s) \\
   = (t \times {t^2} - t \times s) + ({8^2} \times {t^2} - {8^2} \times s) \\
   = ({t^3} - st) + (64{t^2} - 64s) \\
   = {t^3} - st + 64{t^2} - 64s \\
 \]
Hence the simplified expression is \[{t^3} - st + 64{t^2} - 64s\].
(iv) Consider the expression \[(a + b)(c - d) + (a - b)(c + d) + 2(ac + bd)\]
\[
  (a + b)(c - d) + (a - b)(c + d) + 2(ac + bd) \\
   = [a(c - d) + b(c - d)] + [a(c + d) - b(c + d)] + (2 \times ac + 2 \times bd) \\
   = (a \times c - a \times d) + (b \times c - b \times d) + (a \times c + a \times d) - (b \times c + b \times d) + (2 \times ac + 2 \times bd) \\
   = (ac - ad) + (bc - bd) + (ac + ad) - (bc + bd) + (2ac + 2bd) \\
   = ac - ad + bc - bd + ac + ad - bc - bd + 2ac + 2bd \\
   = \underline {ac + ac + 2ac} - ad + ad + bc - bc - bd - bd + 2bd \\
   = 3ac\underline { - ad + ad} + bc - bc - bd - bd + 2bd \\
   = 3ac\underline { + bc - bc} - bd - bd + 2bd \\
   = 3ac\underline { - bd - bd + 2bd} \\
   = 3ac \\
 \]
As like terms with different signs cancel out each other, we are left with\[3ac\].
Hence the simplified expression is\[3ac\].
(v) Consider the expression \[(x + y)(2x + y) + (x + 2y)(x - y)\]
(vi) Consider the expression \[(x + y)({x^2} - xy + {y^2})\]
\[
  (x + y)({x^2} - xy + {y^2}) = x({x^2} - xy + {y^2}) + y({x^2} - xy + {y^2}) \\
   = (x \times {x^2} - x \times xy + x \times {y^2}) + (y \times {x^2} - y \times xy + y \times {y^2}) \\
   = ({x^3} - {x^2}y + x{y^2}) + ({x^2}y - x{y^2} + {y^3}) \\
   = {x^3} - {x^2}y + x{y^2} + {x^2}y - x{y^2} + {y^3} \\
   = {x^3}\underline { - {x^2}y + {x^2}y} + \underline {x{y^2} - x{y^2}} + {y^3} \\
   = {x^3} + {y^3} \\
 \]
Hence the simplified expression is \[{x^3} + {y^3}\].
(vii) Consider the expression \[(1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y\]
\[
  (1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y \\
   = 1.5x(1.5x + 4y + 3) - 4y(1.5x + 4y + 3) - 4.5x + 12y \\
   = (1.5x \times 1.5x + 1.5x \times 4y + 1.5x \times 3) - (4y \times 1.5x + 4y \times 4y + 4y \times 3) - 4.5x + 12y \\
   = (2.25{x^2} + 6xy + 4.5x) - (6xy + 16{y^2} + 12y) - 4.5x + 12y \\
   = 2.25{x^2} + 6xy + 4.5x - 6xy - 16{y^2} - 12y - 4.5x + 12y \\
   = 2.25{x^2} + \underline {6xy - 6xy} + \underline {4.5x - 4.5x} - 16{y^2}\underline { - 12y + 12y} \\
   = 2.25{x^2} - 16{y^2} \\
 \]
Hence the simplified expression is \[2.25{x^2} - 16{y^2}\].
(viii) Consider the expression \[(a + b + c)(a + b - c)\]
\[
  (a + b + c)(a + b - c) = a(a + b - c) + b(a + b - c) + c(a + b - c) \\
   = (a \times a + a \times b - a \times c) + (b \times a + b \times b - b \times c) + (c \times a + c \times b - c \times c) \\
   = \,({a^2} + ab - ac) + (ab + {b^2} - bc) + (ca + bc - {c^2}) \\
   = {a^2} + ab - ac + ab + {b^2} - bc + ca + bc - {c^2} \\
   = {a^2} + \underline {ab + ab} - \underline {ac + ca} + {b^2}\underline { - bc + bc} - {c^2} \\
   = {a^2} + 2ab + {b^2} - {c^2} \\
 \]
Hence the simplified expression is \[{a^2} + 2ab + {b^2} - {c^2}\].

Note: If there is a minus sign outside the bracket, then while opening the bracket, the signs of the terms inside the bracket change.
For example, \[(25 + 6x) - (16x + 20{x^2}) = 25 + 6x - 16x - 20{x^2}\]