STARTER – RANDOM REVISION! JUNE 2010 EXAM Q

MARK SCHEME ANNOTATE & CORRECT IN COLOUR! FINAL MARKS TO YOUR C1 PAST PAPER CHART UNDER 1 ST GO

FACTOR THEOREM AS Maths with Liz Core 1

WHAT IS THE FACTOR THEOREM? Using your notes from last week’s HW, can you explain this process in plain English? This means that we can tell if a given value is a solution/root if we get zero when we substitute it into the function. If a given value, let’s call it a, is a solution, then we know that (x – a) has to be a factor.

EXAMPLE 1 Let’s explore further by looking at the graph of. Cubics form an s-shape curve If leading coefficient is positive, the end- behaviour on the L.H.S. will be facing downwards, and the R.H.S. will be facing upwards. Can you sketch a cubic function with a negative leading coefficient? The roots occur when y=0, so in this case the solutions are x = -3, x = -1, x = 2 Thus, our factors must be (x+3)(x+1)(x-2). Let’s prove this using the factor theorem!

EXAMPLE 1 Let’s explore further by looking at the graph of. Thus, our factors must be (x+3)(x+1)(x-2). Let’s prove this using the factor theorem! Usually you won’t have any information except the given function to start with: If we factorise, we get, where (once you expand brackets). So we need to check and see which factors of -6 equal zero if substituted into the expression. List the factors of -6…

EXAMPLE 1 Let’s explore further by looking at the graph of. Thus, our factors must be (x+3)(x+1)(x-2). Let’s prove this using the factor theorem! Check each value to see if it is a factor: From here, we can keep checking every value until we find our 3 rd solution, OR we can use logic and find the answer much quicker! So far we know that How can we find r?

EXAMPLE 1 Let’s explore further by looking at the graph of. Thus, our factors must be (x+3)(x+1)(x-2). Let’s prove this using the factor theorem! From here, we can keep checking every value until we find our 3 rd solution, OR we can use logic and find the answer much quicker! So far we know that How can we find r? Compare the constants in the brackets to the constant in the expanded form: This matches the sketch we started with!

YOU TRY! Given the curve (a) Express f(x) as a product of three linear factors. (b) List the roots. (c) List the y-intercept. (d) Sketch the graph and label all intersections with the coordinate axes.

SOLUTION: (a) f(x) = (x – 1) (x – 2) (x + 3) (b) roots occur at x = 1, x = 2, and x = -3 (c) y-intercept occurs at (0, 6) (d)

EXAMPLE 2 This means that the other roots must not be integers. Multiply out the R.H.S. and compare to given polynomial.

EXAMPLE 2

JUNE 2011 EXAM Q Find the remainder when p(x) is divided by x – 3.

MARK SCHEME – JUNE 2011