Simplify and factorize the following algebraic expressions: −9x(x2 - 3) − 5x(2x2 - 5), 3(2a − 7)2, (3x…

Question

Simplify and factorize the following algebraic expressions: −9x(x2 - 3) − 5x(2x2 - 5), 3(2a − 7)2, (3x − 3)(2x + 2), and −3t2 - 15.

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QUESTION 1

1.1 Simplify the following

1.1.1 $-9x(x^2 - 3) - 5x(2x^2 - 5)$

Distribute $-9x$ into the first parenthesis and $-5x$ into the second parenthesis.
$= -9x \cdot x^2 - 9x \cdot (-3) - 5x \cdot 2x^2 - 5x \cdot (-5)$ $= -9x^3 + 27x - 10x^3 + 25x$
Combine like terms.
$= (-9x^3 - 10x^3) + (27x + 25x)$ $= -19x^3 + 52x$
The simplified expression is $\boxed{-19x^3 + 52x}$ .

1.1.2 $3(2a - 7)^2$
Expand the squared term $(2a - 7)^2$ using the formula $(A-B)^2 = A^2 - 2AB + B^2$ .
$(2a - 7)^2 = (2a)^2 - 2(2a)(7) + (7)^2$ $= 4a^2 - 28a + 49$
Multiply the result by 3.
$= 3(4a^2 - 28a + 49)$ $= 3 \cdot 4a^2 - 3 \cdot 28a + 3 \cdot 49$ $= 12a^2 - 84a + 147$
The simplified expression is $\boxed{12a^2 - 84a + 147}$ .

1.1.3 $(3x - 3)(2x + 2)$
Use the FOIL method (First, Outer, Inner, Last) or distributive property to multiply the binomials.
$= (3x)(2x) + (3x)(2) + (-3)(2x) + (-3)(2)$ $= 6x^2 + 6x - 6x - 6$
Combine like terms.
$= 6x^2 + (6x - 6x) - 6$ $= 6x^2 - 6$
The simplified expression is $\boxed{6x^2 - 6}$ .

1.2 Factorise the following

1.2.1 $-3t^2 - 15$
Find the greatest common factor (GCF) of $-3t^2$ and $-15$ .
The GCF is $-3$ .
Factor out the GCF.
$= -3(t^2 + 5)$
The factorised expression is $\boxed{-3(t^2 + 5)}$ .

1.2.2 $5x(2x + 1) - 5(2x + 1)$
Identify the common binomial factor, which is $(2x + 1)$ .
Factor out the common binomial.
$= (2x + 1)(5x - 5)$
Notice that $5x - 5$ has a common factor of 5.
$= 5(2x + 1)(x - 1)$
The factorised expression is $\boxed{5(2x + 1)(x - 1)}$ .

1.2.3 $2x(y - p) - 5d(2x + 1)$
This expression does not have a common factor between the two terms $2x(y-p)$ and $5d(2x+1)$ . Therefore, it cannot be factored further in a simple way.
The expression is $\boxed{2x(y - p) - 5d(2x + 1)}$ .

1.2.4 $81 - y^2$
Recognize this as a difference of squares, $A^2 - B^2 = (A - B)(A + B)$ , where $A = \sqrt{81} = 9$ and $B = \sqrt{y^2} = y$ .
$= (9 - y)(9 + y)$
The factorised expression is $\boxed{(9 - y)(9 + y)}$ .

QUESTION 2

2.1 Simplify the following

2.1.1 $\frac{10x^4 + 15x^2}{-5x}$
Divide each term in the numerator by the denominator.
$= \frac{10x^4}{-5x} + \frac{15x^2}{-5x}$
Simplify each term.
$= -2x^{4-1} - 3x^{2-1}$ $= -2x^3 - 3x$
The simplified expression is $\boxed{-2x^3 - 3x}$ .

2.1.2 $\frac{3xy + y^2}{x - 2}$
Factor out the common factor $y$ from the numerator.
$= \frac{y(3x + y)}{x - 2}$
There are no common factors between the numerator and the denominator to cancel.
The simplified expression is $\boxed{\frac{y(3x + y)}{x - 2}}$ .

2.1.3 $\frac{x^2 + 2x - 8}{x - 2}$
Factor the quadratic expression in the numerator $x^2 + 2x - 8$ .
We need two numbers that multiply to -8 and add to 2. These numbers are 4 and -2.
$x^2 + 2x - 8 = (x + 4)(x - 2)$
Substitute the factored numerator back into the expression.
$= \frac{(x + 4)(x - 2)}{x - 2}$
Cancel the common factor $(x - 2)$ , assuming $x \neq 2$ .
$= x + 4$
The simplified expression is $\boxed{x + 4}$ .

2.2 Solve for x

2.2.1 $x - 19 = 14$
Add 19 to both sides of the equation.
$x - 19 + 19 = 14 + 19$ $x = 33$
The solution is $\boxed{x = 33}$ .

2.2.2 $3x - 10 = -22$
Add 10 to both sides of the equation.
$3x - 10 + 10 = -22 + 10$ $3x = -12$
Divide both sides by 3.
$\frac{3x}{3} = \frac{-12}{3}$ $x = -4$
The solution is $\boxed{x = -4}$ .

2.2.3 $x^2 + 9x = -14$
Move all terms to one side to set the equation to zero.
$x^2 + 9x + 14 = 0$
Factor the quadratic expression.
We need two numbers that multiply to 14 and add to 9. These numbers are 2 and 7.
$(x + 2)(x + 7) = 0$
Set each factor equal to zero and solve for $x$ .
$x + 2 = 0 \quad \text{or} \quad x + 7 = 0$ $x = -2 \quad \text{or} \quad x = -7$
The solutions are $\boxed{x = -2 \text{ or } x = -7}$ .

QUESTION 3

3.1 Say whether the following graphs are (increasing, decreasing or constant)

3.1.1 The graph shows a line sloping downwards from left to right.
This graph is decreasing.

3.1.2 The graph shows a line sloping upwards from left to right.
This graph is increasing.

3.1.3 The graph shows a horizontal line.
This graph is constant.

Simplify and factorize the following algebraic expressions: -9x(x2 - 3) - 5x(2x2 - 5), 3(2a - 7)2, (3x - 3)(2x + 2), and -3t2 - 15.

Step-by-step solution

Distribute $-9x$ into the first parenthesis and $-5x$ into the second parenthesis.

Combine like terms.

Expand the squared term $(2a - 7)^2$ using the formula $(A-B)^2 = A^2 - 2AB + B^2$ .

Multiply the result by 3.

Use the FOIL method (First, Outer, Inner, Last) or distributive property to multiply the binomials.

Combine like terms.

Find the greatest common factor (GCF) of $-3t^2$ and $-15$ .

Factor out the GCF.

Identify the common binomial factor, which is $(2x + 1)$ .

Factor out the common binomial.

Notice that $5x - 5$ has a common factor of 5.

Recognize this as a difference of squares, $A^2 - B^2 = (A - B)(A + B)$ , where $A = \sqrt{81} = 9$ and $B = \sqrt{y^2} = y$ .

Divide each term in the numerator by the denominator.

Simplify each term.

Factor out the common factor $y$ from the numerator.

Factor the quadratic expression in the numerator $x^2 + 2x - 8$ .

Substitute the factored numerator back into the expression.

Cancel the common factor $(x - 2)$ , assuming $x \neq 2$ .

Add 19 to both sides of the equation.

Add 10 to both sides of the equation.

Divide both sides by 3.

Move all terms to one side to set the equation to zero.

Factor the quadratic expression.

Set each factor equal to zero and solve for $x$ .

Still stuck on this one?

More Mathematics Questions

Simplify and factorize the following algebraic expressions: -9x(x2 - 3) - 5x(2x2 - 5), 3(2a - 7)2, (3x - 3)(2x + 2), and -3t2 - 15.

Step-by-step solution

Distribute −9x-9x−9x into the first parenthesis and −5x-5x−5x into the second parenthesis.

Combine like terms.

Expand the squared term (2a−7)2(2a - 7)^2(2a−7)2 using the formula (A−B)2=A2−2AB+B2(A-B)^2 = A^2 - 2AB + B^2(A−B)2=A2−2AB+B2.

Multiply the result by 3.

Use the FOIL method (First, Outer, Inner, Last) or distributive property to multiply the binomials.

Combine like terms.

Find the greatest common factor (GCF) of −3t2-3t^2−3t2 and −15-15−15.

Factor out the GCF.

Identify the common binomial factor, which is (2x+1)(2x + 1)(2x+1).

Factor out the common binomial.

Notice that 5x−55x - 55x−5 has a common factor of 5.

Recognize this as a difference of squares, A2−B2=(A−B)(A+B)A^2 - B^2 = (A - B)(A + B)A2−B2=(A−B)(A+B), where A=81=9A = \sqrt{81} = 9A=81​=9 and B=y2=yB = \sqrt{y^2} = yB=y2​=y.

Divide each term in the numerator by the denominator.

Simplify each term.

Factor out the common factor yyy from the numerator.

Factor the quadratic expression in the numerator x2+2x−8x^2 + 2x - 8x2+2x−8.

Substitute the factored numerator back into the expression.

Cancel the common factor (x−2)(x - 2)(x−2), assuming x≠2x \neq 2x=2.

Add 19 to both sides of the equation.

Add 10 to both sides of the equation.

Divide both sides by 3.

Move all terms to one side to set the equation to zero.

Factor the quadratic expression.

Set each factor equal to zero and solve for xxx.

Still stuck on this one?

More Mathematics Questions

Distribute $-9x$ into the first parenthesis and $-5x$ into the second parenthesis.

Expand the squared term $(2a - 7)^2$ using the formula $(A-B)^2 = A^2 - 2AB + B^2$ .

Find the greatest common factor (GCF) of $-3t^2$ and $-15$ .

Identify the common binomial factor, which is $(2x + 1)$ .

Notice that $5x - 5$ has a common factor of 5.

Recognize this as a difference of squares, $A^2 - B^2 = (A - B)(A + B)$ , where $A = \sqrt{81} = 9$ and $B = \sqrt{y^2} = y$ .

Factor out the common factor $y$ from the numerator.

Factor the quadratic expression in the numerator $x^2 + 2x - 8$ .

Cancel the common factor $(x - 2)$ , assuming $x \neq 2$ .

Set each factor equal to zero and solve for $x$ .