[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
\n” _builder_version=”4.11.3″ _module_preset=”default” header_font=”|700|||||||” header_text_align=”left” header_font_size=”50px” header_line_height=”1.18em” custom_padding=”|0px||4px|false|false” header_font_size_tablet=”” header_font_size_phone=”35px” header_font_size_last_edited=”on|phone” global_colors_info=”{}”]<\/p>\n
<\/p>\n
[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” hover_enabled=”0″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n
Factorisation is a method that helps to find the factors of numbers or mathematical expressions. It is defined as dividing or breaking an entity into a product of several smaller entities which are of the same type, i.e., numbers, algebraic terms, etc.<\/span><\/p>\n <\/span><\/p>\n Any number is a factor of a number if it divides the number without leaving any remainder behind.\u00a0For example, 2 is a factor of 4, 5 is a factor of 20 hence, the factorisation of 20 = 4 \u00d7 5.<\/span><\/p>\n <\/span><\/p>\n Now that we know what factorisation is, so it is easier to understand the factorisation of algebraic expressions. In this case, the expression is reduced to a simpler form, and are represented as a product of its factors. The factors of an algebraic expression are either integers, variables, or a simpler algebraic expression.<\/span><\/p>\n For example, the factorisation of the expression 3 x^{2}+9 x-30 \\text { is }<\/span><\/span><\/p>\n 3 x^{2}+9 x-30=3(x+5)(x-2)<\/span>. <\/span>Here, 3, (x+5)\\text{ and }(x-2)<\/span>\u00a0 are the factors of the expression.<\/span><\/p>\n <\/span><\/p>\n There are three ways using which an expression can be factorised, these are:<\/span><\/p>\n Let\u2019s discuss each method in detail.<\/span><\/p>\n <\/span><\/p>\n Let us consider an algebraic expression:<\/span><\/p>\n y^{3}+2 y^{2}<\/span><\/span><\/p>\n Here y^{3} \\text { can be factorised as } y \\times y \\times y<\/span>.<\/span><\/p>\n 2 y^{2} \\text { can be factorised as } 2 \\times y \\times y \\text {. }<\/span><\/span><\/p>\n The Greatest common factor of the two terms is y^{2}<\/span><\/span>.<\/span><\/p>\n Hence, the factorised expression is y^{2}(y+2)<\/span><\/span>.<\/span><\/p>\n This is the common factor method.<\/span><\/p>\n <\/span><\/p>\n For some cases all the terms in the expression may not have a single common factor, here the regrouping method comes to use. Let us consider an algebraic expression ( 28x + z + 2xz + 14 ).<\/span><\/p>\n Identify which terms have common factors. Here the first and the last term has a common factor 14. The second and third terms have a common factor z.<\/span><\/p>\n Thus, we regroup the terms:<\/span><\/p>\n 28x + z + 2xz + 14 = ( 28x + 14 ) + ( z + 2xz )\u00a0<\/span><\/p>\n Now following the method of common factor:<\/span><\/p>\n ( 28x + 14 ) + ( z + 2xz ) = 14( 2x + 1 ) + z( 2x + 1 ).<\/span><\/p>\n Clearly ( 2x + 1 ) is a common factor, hence the factorised expression is:<\/span><\/p>\n ( 2x + 1 )( 14 + z ) which can be written as ( 2x + 1 )( z + 14 ).<\/span><\/p>\n <\/span><\/p>\n List of common Identities<\/b><\/p>\n The list of common identities that are used for factorisation is:<\/span><\/p>\n 1. (a+b)^{2}=a^{2}+2 a b+b^{2}<\/span><\/span><\/p>\n 2. (a-b)^{2}=a^{2}-2 a b+b^{2}<\/span><\/span><\/p>\n 3. a^{2}-b^{2}=(a+b)(a-b)<\/span><\/span><\/p>\n 4. (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}<\/span><\/span><\/p>\n 5. (a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}<\/span><\/span><\/p>\n 6. a^{3}+b^{3}=(a+b)\\left(a^{2}-a b+b^{2}\\right)<\/span><\/span><\/p>\n 7. a^{3}-b^{3}=(a-b)\\left(a^{2}+a b+b^{2}\\right)<\/span><\/span><\/p>\n <\/span><\/p>\n For example, 4 y^{2}+4 y+1<\/span> <\/span>is an algebraic expression, however, there are no common factors for the three terms in the expression. In this case, we check if the expression is similar to an identity to factorize easily.<\/span><\/p>\n <\/span><\/p>\n 4 y^{2}+4 y+1=(2 y)^{2}+2 \\cdot(2 y) \\cdot 1+1^{2}<\/span><\/span><\/p>\n Which is similar to the expression\u00a0 (a+b)^{2}=a^{2}+2 a b+b^{2}<\/span>.<\/span><\/p>\n Comparing the two expressions gives us the factors of the expression 4 y^{2}+4 y+1<\/span>.<\/span><\/p>\n 4 y^{2}+4 y+1=(2 y+1)^{2}=(2 y+1)(2 y+1)<\/span><\/span><\/p>\n \\therefore \\text { the factor is }(2 y+1)^{2} \\text { or the factors are }(2 y+1) \\text { and }(2 y+1) \\text {. }<\/span><\/span><\/p>\n <\/span><\/p>\n Example 1:<\/strong> Determine the factors of the algebraic expression a^{4}+6 a^{3}+9 a^{2}<\/span><\/span><\/b><\/p>\n Solution<\/strong>:<\/strong>\u00a0<\/span><\/p>\n The given expression is a^{4}+6 a^{3}+9 a^{2}<\/span>.<\/span><\/p>\n The common factor in all the three terms is a^2<\/span><\/span>, hence the expression reduces to:<\/span><\/p>\n a^{4}+6 a^{3}+9 a^{2}=a^{2}\\left(a^{2}+6 a+9\\right)<\/span><\/span><\/p>\n The part in the parenthesis can be factorised further using identity:<\/span><\/p>\n (x+b)^{2}=x^{2}+2 x b+b^{2}<\/span><\/span><\/p>\n a^{2}+6 a+9=a^{2}+2 \\cdot(a) \\cdot(3)+(3)^{2}=(a+3)^{2}<\/span><\/span><\/p>\n a^{4}+6 a^{3}+9 a^{2}=a^{2}(a+3)^{2}<\/span><\/span><\/p>\n The factors of the expression are a, a, (a+3),(a+3)<\/span>.<\/span><\/p>\n <\/span><\/p>\n Example 2<\/strong>:<\/strong> Are 2, ( z + 2 ), ( z \u2013 3 ) the factors of 2 z^{2}-2 z-12 ?<\/span><\/span><\/p>\n Solution:<\/strong><\/p>\n The given terms are 2,(z+2) \\text { and }(z-3)<\/span>. To check if they are the factors or not we must calculate the product of the terms.<\/span><\/p>\n 2 \\times(z+2) \\times(z-3)=2[(z+2)(z-3)]=2\\left[z^{2}-3 z+2 z-6\\right]<\/span><\/span><\/p>\n \\text { Now, } 2\\left[z^{2}-3 z+2 z-6\\right]=2\\left[z^{2}-z-6\\right]=2 z^{2}-2 z-12<\/span><\/span><\/p>\n \\text { Hence, } 2,(z+2) \\text { and }(z-3) \\text { are the factors of } 2 z^{2}-2 z-12 \\text {. }<\/span><\/span><\/p>\n [\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\nFactorisation of Numbers<\/b><\/h2>\n
Factorisation of Algebraic Expressions<\/b><\/h2>\n
Types of Factorisations<\/b><\/b><\/h2>\n
\n
Common Factor Method<\/b><\/h3>\n
Regrouping Factor Method<\/b><\/h3>\n
Factoring using Identities<\/b><\/h3>\n
Examples<\/b><\/h2>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nExplore Other Topics<\/h1>\n