Q1. Simplify Daily Practice 28.6.2016 Q2. Evaluate Today we will be learning about Polynomials. Q3. Write in completed square form x 2 + 4x + 7 Q4. State the equation of the line joining (0, 3) and (4, -6) Q5. Factorise 6x 2 + 17x + 12 Polynomials Polynomials are expressions that have x with a power in them. The degree of the polynomial is the value of the highest power. E.g. f(x) = x 2 + 3x is a quadratic, f(x) = 3x 3 + x 2 is a cubic etc. Synthetic Division A polynomial can be continually factorised which is known as nested form. This form is used as a division method. If f(h) = ah 4 + bh 3 + ch 2 + dh + e A number in front of an x n is called a coefficient. A root of a polynomial is the value of x for which f(x) or y = 0 Synthetic Division: Factorising Polynomials This is a method used instead of long division for dividing a polynomial by a factor and finding the remainder if there is one. Synthetic Division: Dividing by (x - h) We will be dividing by one of the roots. Therefore if dividing by (x - a) the root is x = a. Before dividing, the polynomial should be arranged in descending powers of x and if there are any missing powers, use 0 as the coefficient. 1. Divide x 3 - x 2 + 3x - 4 by x - 2, and write down the quotient and the remainder. E.g. f(x) = 4x 2 + 3x 4 + 2x + 1 should be written as 3x 4 + 0x 3 + 4x 2 + 2x + 1
Daily Practice 17.8.2016 Q1. State the gradient of the equation 3x + 4y - 7 = 0 Q2. Given U n+1 = 0.8U n + 4, where U 0 = 12, (i) Find the values of U 2 and U 3 (ii) State the limit of the above sequence Today we will be learning how to factorise polynomials using synthetic division. Q3. Find the range of values of x for which x 2 + 4x - 12 > 0 Synthetic Division: Dividing by (x - h) 2. Divide x 3 + 6x 2 + 3x - 15 by x + 3, and write down the quotient and the remainder. Synthetic Division: Dividing by (x - h) 3. Divide x 4 - x 2 + 7 by x + 1, and write down the quotient and the remainder. Daily Practice 18.8.2016 Q1. Write 5x 2-10x + 4 in the form a(x + p) 2 + q Q2. State the gradient of the line 3x + 4y = 7 Today we will be continuing synthetic division. Q3. State the limit of the recurrence relation u n+1 = 0.4u n + 5 Q4. State the nature of the roots of the function f(x) = 3x 2 + 2x - 5
Synthetic Division: Dividing by (ax - b) 2x 3 + x 2 + 5x - 1 (2x - 1) Synthetic Division: Dividing by (ax - b) 2. 25x 3 + 11x + 4 (5x + 2) The Factor Theorem If a polynomial is divided by (ax - h) and the remainder is zero, then (ax - h) must be a factor of the polynomial. Conversely, if (x - h) is a factor of f(x), then f(h) = 0. i.e. the remainder is zero. The Factor Theorem 1. Show that (x - 2) is a factor of f(x) = 2x 3-11x 2 + 17x - 6 and hence factorise fully the polynomial. The Factor Theorem 2. Factorise fully, the polynomial x 3 - x 2-14x + 24 Daily Practice 19.8.2016 1. Show that the line y = 5x - 2 is a tangent to the curve y = 2x 2 + x and find the point of contact 2. Write the function f(x) = 3x 2 + 9x - 1 in the form a(x + p) 2 + q 3. If the graph of y = kx 2-3x + 2 cuts the x - axis in two places, what are the range of values of k?
The Factor Theorem: Finding Coefficients 1. Find p if (x + 2) is a factor of x 3 + 6x 2 + px + 6 Today we will be learning how to find coefficients using the factor theorem. The Factor Theorem: Finding Coefficients 2. The Factor Theorem: Finding Coefficients 3. Find the values of 'a' and 'b' if (x - 3) and (x + 2) are factors of x 3 + ax 2 + bx + 42. Hence factorise fully the polynomial. Ex. 7F Q1. (b) (c) Q2 (b) Q3, Q5. Daily Practice 22.8.2016 Q1. Find the value of k such that the equation kx 2 + kx + 6 has equal roots Q2. Given that x + 2 is a factor of 2x 3 + x 2 + kx + 2, find the value of k Q3. Show that the line y = 2x + 1 does not intersect with the parabola with equation y = x 2 + 3x + 4 Today we will be learning how to solve polynomial equations. Daily Question Online from Today
Q1. Daily Practice 23.8.2016 Q2. Today we will be learning to solve polynomials. Check Daily Question. Solving Polynomial Equations Similar to solving quadratic equations. If required, rearrange the equation to make it equal zero Find a factor and then factorise by using synthetic division Make each factor equal zero State the roots Solving Polynomial Equations Solving Polynomial Equations 1. Solve x 3 + 6x 2 + 3x - 10 = 0
1. Solve 2x 3-3x 2-18x + 27 = 0 Daily Practice 24.8.2016 Q1. Q2. 5. Factorising and solving a quartic Example: Solve x 4 + 4x 3-17x 2-24x + 36 = 0 Today we will be learning how to state the equation of a polynomial given its graph. Stating the equation given its graph f(x) = k(x - a)(x - b)(x - c) is the general equation of a polynomial. a, b, and c are the roots and you can find k by substituting the y - intercept or another point on the graph. y 1. Using the graph shown, find an expression for f(x) (0,12) - 4-1 0 1 x
Stating the equation given its graph 2. Using the graph shown, find an expression for f(x) y The T.P. on the x - axis represents a repeated root -1 0 2 x (1,-4)