. I Simplify the following algebraie expressions i. Usc rule of distribution lu expand (14x-5)(3x-4) ii. Use identities rule to the expression (4cdot x)^2 2a^3b^7times 2ab^5 iv. (14x-5)(3x-4) (9x-11)(3x+8) 2. A Polynomial (x) is multiple of x-3 and the remainder when P(x) is divided by x+3 is Determine the remainder when P(x) is divided by x^a-9 is marks) 3 . Applying splitting the middle term method factorise the expression 35x^2-54x+16 (3)marks) (5 marks) 4. Establish relations between log_(a)x=p and a^p=x 5. Use matrix method to solve the system of simultaneous equations below (5 marks) x-4y=3 3x-y=-2 b. State properties of matrix multiplication 7. If A=(} 1&2 -2&3 ) Verify that A(B+C)=AB+AC ifB+C= ) -1&2 4&3 8.Use first principle to differentiate the function y=(1)/(x^3) (5 marks) cs marks)

1. Simplify the following algebraic expressions:<br /> i. $(14x-5)(3x-4)$<br /> - Using the distributive property, we get:<br /> $(14x-5)(3x-4) = 42x^2 - 56x - 15x + 20 = 42x^2 - 71x + 20$<br /><br /> ii. $(4\cdot x)^{2}$<br /> - Using the power of a product rule, we get:<br /> $(4\cdot x)^{2} = 16x^2$<br /><br /> iii. $2a^{3}b^{7}\times 2ab^{5}$<br /> - Using the product of powers rule, we get:<br /> $2a^{3}b^{7}\times 2ab^{5} = 4a^{4}b^{12}$<br /><br /> iv. $(14x-5)(3x-4)$<br /> - This is the same as the first expression, so the answer is the same:<br /> $(14x-5)(3x-4) = 42x^2 - 71x + 20$<br /><br /> v. $(9x-11)(3x+8)$<br /> - Using the distributive property, we get:<br /> $(9x-11)(3x+8) = 27x^2 + 72x - 33x - 88 = 27x^2 + 39x - 88$<br /><br />2. A Polynomial $P(x)$ is multiple of $x-3$ and the remainder when $P(x)$ is divided by $x+3$ is $5$. Determine the remainder when $P(x)$ is divided by $x^{2}-9$.<br /> - Since $P(x)$ is a multiple of $x-3$, we can write $P(x) = (x-3)Q(x)$, where $Q(x)$ is some polynomial.<br /> - When $P(x)$ is divided by $x+3$, the remainder is $5$, so we can write $P(x) = (x+3)R(x) + 5$, where $R(x)$ is some polynomial.<br /> - Since $P(x)$ is a multiple of $x-3$, we can write $P(x) = (x-3)Q(x) = (x+3)R(x) + 5$.<br /> - Equating the coefficients of $x$, we get $Q(x) = R(x) + \frac{5}{x-3}$.<br /> - Substituting $x = 3$, we get $Q(3) = R(3) + \frac{5}{3-3} = R(3) + \infty$, which is not possible.<br /> - Therefore, the remainder when $P(x)$ is divided by $x^{2}-9$ is $5$.<br /><br />3. Applying splitting the middle term method factorise the expression $35x^{2}-54x+16$.<br /> - We can write the expression as $35x^{2} - 28x - 26x + 16$.<br /> - Grouping the terms, we get $(35x^{2} - 28x) - (26x - 16)$.<br /> - Factoring out the common factors, we get $7x(5x - 4) - 2(13x - 8)$.<br /> - Rearranging the terms, we get $(7x - 2)(5x - 4)$.<br /><br />4. Establish relations between $log_{a}x=p$ and $a^{p}=x$.<br /> - If $log_{a}x=p$, then $a^{p}=x$.<br /> - Conversely, if $a^{p}=x$, then $log_{a}x=p$.<br /><br />5. Use matrix method to solve the system of simultaneous equations below:<br /> $x-4y=3$<br /> $3x-y=-2$<br /> - Representing the system of equations in matrix form, we get:<br /> $\begin{bmatrix} 1 & -4 \\ 3 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3 \\ -2 \end{bmatrix}$<br /> - Solving the system using the inverse matrix method, we get:<br /> $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 & -4 \\ 3 & -1 \end{bmatrix}^{-1} \begin{bmatrix} 3 \\ -2 \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$<br /><br />6. State properties of matrix multiplication.<br /> - Matrix multiplication is associative: $A(BC) = (AB)C$<br /> - Matrix multiplication is not commutative: $AB \neq BA$<br /> - The product of a matrix